inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $X_1, X_2, \ldots, X_n$ be a random sample from a continuous distribution with unknown mean $\mu$ and known standard deviation $\sigma$. If $[X_{\min}, X_{\max}]$ is used as a confidence interval for $\mu$, determine the confidence level in terms of the standard normal distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - 2[\Phi(0)]^n}
Find the set $T$ of complex numbers defined by \[ T = \left\{ z \in \mathbb{C} \mid \left|\frac{z - i}{z + i}\right| = 1 \right\}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Find the residue of the sum \( S = 1^{965}+2^{965}+\cdots+2018^{965} \) when divided by \(2021\). Express your answer as \(S \equiv x \ (\text{mod}\ 2021)\). What is the value of \(x\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
Determine the completion of the metric space \(c_{00}\), defined as \(c_{00} = \{(x_n) \in \mathbb{R}: x_n=0, n>n_0\}\), with the metric induced from \(l_{\infty}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c_0}
Determine the degree of nilpotency of the dihedral group $G = D_{16}$ using only the upper central series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
If the 26 letters of the alphabet are arranged in a circle such that no two vowels are adjacent, what is the probability that the letters 'a' and 'b' are next to each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{21}}
Consider a string of \( {n7’}\mathrm{\;s},{7777}\cdots {77} \) ,into which+signs are inserted to produce an arithmetic expression. For example,
 \( 7 + {77} + {777} + 7 + 7 = {875} \) could be obtained from eight 7’s in this way. For how many values of \( n \) is it possible to insert+
 signs so that the resulting expression has value 7000 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
108
}
Consider a string of $n$ 1's. We wish to place some $+$ signs in between so that the sum is $1000$. For instance, if $n=190$, one may put $+$ signs so as to get $11$ ninety times and $1$ ten times, and get the sum $1000$. If $a$ is the number of positive integers $n$ for which it is possible to place $+$ signs so as to get the sum $1000$, then find the sum of digits of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Calculate the residue of the function \( \frac{z^{2}}{(z+i)^{2}(z-i)^{2}} \) at the pole \( z = i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{i}{4}}
What is the maximum number of distinct integers in a row such that the sum of any 11 consequent integers is either 100 or 101 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
22
}
There is a pile of 10,000,000 matchsticks. Two players take turns in a game where on each turn, the current player can take away \( P^n \) matchsticks from the pile, where \( P \) is a prime number and \( n = 0,1,2,3, \cdots \) (for example, the first player might take 25 matchsticks, the second player 8 matchsticks, then the first player 1 matchstick, the second player 5 matchsticks, and so on). The player who takes the last matchstick wins the game. In a game played correctly, who will win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{First Player}
Given the cumulative distribution function (CDF) of a random variable X:

\[ F(x) = \begin{cases} 0, & \text{if } x < 0 \\ \frac{1}{4}, & \text{if } 0 \le x < 1 \\ \frac{2}{5}, & \text{if } 1 \le x < 2 \\ \frac{2x-3}{2}, & \text{if } 2 \le x < 2.5 \\ 1, & \text{if } x \ge 2.5 \end{cases} \]

Calculate the expected value E(X).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{59}{40}}
Determine the missing term in the equation \( z=\sqrt{r^2-x^2-y^2}\sin(?) \), where the equation represents a 3D standing electron wave with 2 periods of oscillation per rotation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\phi}
Given positive numbers $x$ and $y$ satisfying $x^2+y^2=1$, find the maximum value of $\frac {1}{x}+ \frac {1}{y}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
You have a four-sided die numbered 1 to 4 and are playing a game. In the first round, you roll the die once. If you roll a 1, you lose the die. If you roll a 2, you keep the die. If you roll a 3 or 4, you gain an additional identical die. In the second round, you roll each die you have, applying the same rules to each die. You continue this process in subsequent rounds. If you ever have zero dice, you lose the game. What is the probability that you never lose the game over infinite rounds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
The smaller leg of a rectangular trapezoid is equal to 3, and the larger leg forms an angle of $30^{\circ}$ with one of the bases. Find this base if the point of intersection of the angle bisectors at the other base lies on it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
若 $\triangle A B C$ 的角 $A, C$ 满足 $5(\cos A+\cos C)+4(\cos A \cos C+1)=0$, 则 $\tan \frac{A}{2} \cdot \tan \frac{C}{2}=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Suppose \( \alpha ,\beta ,\gamma  \in  \{  - 2,3\} \) are chosen such that
 \( M = \mathop{\max }\limits_{{x \in  \mathbb{R}}}\mathop{\min }\limits_{{y \in  {\mathbb{R}}_{ \geq  0}}}{\alpha x} + {\beta y} + {\gamma xy} \)
 is finite and positive (note: \( {\mathbb{R}}_{ \ge  0} \) is the set of nonnegative real numbers).
 What is the sum of the possible values of \( M \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{2}}
Given the function \( V(x(t)) = \frac{x''}{x} \), where the prime denotes differentiation with respect to time, find \( \frac{dV(x(t))}{dx} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{x''' x - x'' x'}{x^2 x'}}
Evaluate the limit: \[ \lim_{n \to \infty} \left[ \ln\left(1 - \frac{1}{1^2}\right) + \ln\left(1 - \frac{1}{2^2}\right) + \cdots + \ln\left(1 - \frac{1}{n^2}\right) \right] \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln 2}
Find the fundamental group of the space obtained by removing all points with integer coordinates from three-dimensional Euclidean space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
At the Bertolli Farm, they grow 2073 tomatoes, 4112 cobs of corn, 985 onions, and 6250 carrots. The growth rates for each crop are as follows: 12% for tomatoes, 15% for cobs of corn, 8% for onions, and 10% for carrots. Calculate the difference in total yield between the highest and lowest yielding crops after accounting for their respective growth rates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5811.2}
Evaluate the integral \( \int_{0}^{1}\frac{x^3-1}{\log(x)} \,\mathrm{d}x \) using a method other than the Feynman integration method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 4}
Evaluate the path integral of the vector field \( F = \langle \arccos x, xy - e^y \rangle \) over the triangular path from \((0,0)\) to \((2,3)\) to \((2,0)\) and back to \((0,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the maximal number of elements in a sigma-algebra generated by a collection of N subsets of a set X.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2^N}}
$PQ$ is a diameter of a circle. $PR$ and $QS$ are chords with intersection at $T$. If $\angle PTQ= \theta$, determine the ratio of the area of $\triangle QTP$ to the area of $\triangle SRT$ (i.e. area of $\triangle QTP$/area of $\triangle SRT$) in terms of trigonometric functions of $\theta$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{\cos^2(\theta)}}
Compute the limit: $$ \lim_{z \to 0} \frac{(1-\cos z)^2}{(e^z-1-z)\sin^2z} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
It is winter, and the white rabbit has stored 180 carrots, while the gray rabbit has stored 120 cabbages. The gray rabbit trades some number of cabbages for some of the white rabbit's carrots so that they end up with an equal amount of food. How many carrots can be exchanged for one cabbage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the sum of all the digits of the smallest positive integer \( n \) such that the product \( P_2 \times P_3 \times \ldots \times P_n \) exceeds 2010, where \( P_k = 1 + \frac{1}{k} - \frac{1}{k^2} - \frac{1}{k^3} \) for a positive integer \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Let's determine the value of the following expression:

$$
X=\sqrt{x \sqrt{y \sqrt{z \sqrt{x \sqrt{y \sqrt{z \cdots}}}}}}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[7]{x^{4}y^{2}z}}
Determine the set of values of $k$ for which the polynomial equation \(3x^{4}+4x^3-12x^2+k=0\) has four real roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 5]}
Evaluate the integral \( I = \int_0^\pi \log(\sin(t)) \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi \ln 2}
Determine the mixing time of a discrete random walk on an N-cycle, where the walk moves to the right or left with equal probability of \( \frac{1}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(N^2)}
Evaluate \( \lim_{x \to 0} x^2\left(1+2+3+\ldots+\left[ \frac{1}{|x|} \right] \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Evaluate the integral \( \int_{-\infty}^{+\infty} \frac{\cos(mx)}{x^2+1} \, dx \) using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi e^{-|m|}}
Determine the set of positive values of \( p \) for which the improper integral \( \int_1^{\infty}\frac{e^{-px}}{\ln{x}}dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Convex quadrilateral $ABCD$ has $AB = 18, \angle{A} = 60 \\textdegree$, and $\overline{AB} \parallel \overline{CD}$. In some order, the lengths of the four sides form an arithmetic progression, and side $\overline{AB}$ is a side of maximum length. The length of another side is $a$. What is the sum of all possible values of $a$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84
}
Determine the number of connected components of the set \( S = \{ A \in SL(2, \mathbb{R}) \mid A^T \operatorname{diag}(1, -1) A = \operatorname{diag}(1, -1) \} \) in the induced topology, where \( SL(2, \mathbb{R}) \) is the set of real \( 2 \times 2 \) matrices with determinant 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider a square matrix $A = [a_{ij}] \in \mathbb{R}^{n \times n}$, where each entry is either $1$ or $-1$. If the columns of $A$ are balanced, meaning that the sum of entries in each column is zero, $\sum_{i=1}^n a_{ij} = 0$, determine the maximum possible value of the spectral norm of $A$, denoted as $\|A\|$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
In their base $10$ representation, the integer $a$ consists of a sequence of $1985$ eights and the integer $b$ consists of a sequence of $1985$ fives. Find the sum of the digits of the base 10 representation of $9ab$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17865}
The curves \(y=mx\) and \(y=\frac{x}{x^2+1}\) enclose a region for \(0<m<1\). Find the area of this region in terms of \(m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m - 1 - \ln m}
Given a magician's hat contains 4 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 4 of the reds are drawn or until both green chips are drawn, calculate the probability that all 4 red chips are drawn before both green chips are drawn.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Evaluate the integral \( \displaystyle \int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 3}
Determine the value of $a$ such that the existence and uniqueness theorem applies to the initial value problem: \( x' = \frac{3}{2}|x|^{1/3}, \ x(0) = a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \neq 0}
Suppose $X$ and $Y$ are points on tJhe boundary of the triangular region $ABC$ such that the segment $XY$ divides the region into two parts of equal area. If $XY$ is the shortest such segment and $AB = 5$, $BC = 4$, $AC = 3$ calculate the length of $XY$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Define a sequence $a_i$ as follows: $a_1 = 181$ and for $i \ge 2$, \( a_i = a_{i-1}^2 - 1 \) if $a_{i-1}$ is odd and \( a_i = \frac{a_{i-1}}{2} \) if $a_{i-1}$ is even. Find the least $i$ such that $a_i = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Vasya constructed a robot, but due to some programming mistake, the robot makes random movements, which still somehow lead it to the target point.

A move is a shift of one kilometer either north (N), east (E), south (S), or west (W).

The robot reached its target in 22 moves represented by the sequence:

EENENNWNENESEEEEESSSWN.

How many moves would the robot need if it did not make any unnecessary movements?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate the expression \( \epsilon^{\mu \nu \alpha \beta}\epsilon_{\mu \nu \alpha \beta} \) in four dimensions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-24}
Calculate the limit of the sequence \( c(n)=(n+2)^{(n+2)^{-1/(n+2)}} -(n+1)^{(n+1)^{-1/(n+1)}} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
An ant crawls along the edges of a cube with side length 1 unit. Starting from one of the vertices, the ant travels from one vertex to an adjacent vertex every minute. After 7 minutes, the ant is at a distance of \(\sqrt{3}\) units from the starting point. Determine the number of possible routes the ant could have taken.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{546}
Evaluate the double integral \( \iint\limits_D (x^2-xy-1) \, dx \, dy \), where \( D \) is the triangular region bounded by the lines \( x-2y+2=0, x+3y-3=0 \) and \( y=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{24}}
In the market supply of light bulbs, products from Factory A account for 70%, while those from Factory B account for 30%. The pass rate for Factory A's products is 95%, and the pass rate for Factory B's products is 80%. What is the probability of purchasing a qualified light bulb manufactured by Factory A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.665}
Compute \(d(x_1 x_2)\) given the stochastic differential equations \(dx_1 = a_1 dt + b_1 dw_1(t)\) and \(dx_2 = a_2 dt + b_2 dw_2(t)\), where \(w_1\) and \(w_2\) are independent Wiener processes. Specifically, determine the value of \(d(x_1)d(x_2)\) in the context of Ito calculus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the set of all values of \( p \) such that for every \( q > 0 \), the inequality \( \frac{3(pq^2+p^2q+3q^2+3pq)}{p+q} > 2p^2q \) holds. Express your answer in interval notation in decimal form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 3)}
Some vertices of a regular $2024$-gon are marked such that for any regular polygon, all of whose vertices are vertices of the $2024$-gon, at least one of its vertices is marked. Find the minimal possible number of marked vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{506}
Evaluate the limit \( \lim\limits_{n\to\infty}\int_0^a\frac{x^n} {x^n+1}\,\mathrm dx \), where \( a>0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\max(a - 1, 0)}
Find the correct solution to the integral equation $$f(x) = a - \int^x_b (x-t)f(t)dt$$ where $a$ and $b$ are constants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \cos(x - b)}
Suppose we have \(n\) independent variables from \(U[0,1+\theta]\) and let \(\lambda=1+\theta\). The estimator of \(\theta\) is given by \(\bar{\theta}=2 \bar{X} - 1\). Find the bias of \(\bar{\theta}\) given that the distribution of \(\bar{X}\) for \(n\) variables from \(U[0,1]\) is the Bates distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider a $10 \times 10$ matrix $A$. Each element $a_{ij}$ of the matrix is multiplied by $\frac{i}{j}$, where $1 \leq i,j \leq 10$. Determine how the determinant of the matrix changes after this transformation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Three. (50 points) Given a square table with four legs $L_{1}, L_{2}, L_{3}, L_{4}$, each of length $n$, where $n$ is a positive integer. How many ordered quadruples $\left(k_{1}, k_{2}, k_{3}, k_{4}\right)$ exist such that after sawing off a segment of length $k_{i}$ from the end of leg $L_{i}$ $(i=1,2,3,4)$, the table remains stable? Here, $k_{1}, k_{2}, k_{3}, k_{4}$ are non-negative integers (the table is considered stable if all four legs touch the ground, and it is allowed to saw off a length of 0).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}(n+1)(2n^{2}+4n+3)}
Compute the sum of the squares of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 2 = 0,\] given that all of the roots are real and nonnegative.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Determine the conditions on the constants \(a, b, c\) such that the function \(f(ax+by, cz)\) is harmonic in \(\mathbb{R}^3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^2 + b^2 = c^2}
Six pirates - a captain and five crew members - are sitting around a campfire facing the center. They need to divide a treasure of 180 gold coins. The captain proposes a way to divide the coins (i.e., how many coins each pirate should receive: each pirate must receive a whole non-negative number of coins; different pirates may receive different amounts of coins). After this, the other five pirates vote on the captain's proposal. A pirate will vote "yes" only if he receives more coins than each of his two neighbors. The proposal is accepted if at least three out of the five crew members vote "yes".

What is the maximum number of coins the captain can receive under these rules?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59}
Determine the set of all possible values of the series \( \sum_{n\geq 1}\frac{1}{x_n} \) where \( (x_n)_{n\geq 1} \) is an increasing sequence of positive natural numbers and the series converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \infty)}
A 450 meter long train crosses a man standing on the platform in 8 seconds. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56.25}
Let \( V \) and \( W \) be linear subspaces of \( \mathbb{R}^4 \) defined as follows:

\[ V=\{ \vec{x} \in \mathbb{R}^4 \mid x-y+2z-t=0 \} \]
\[ W=\text{span}(1,1,1,1)^T \]

Find the dimension of the linear subspace \( F=\{ f \in \text{End}(\mathbb{R}^4) \mid f(V) \subseteq W\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Using the digits 1, 2, 3, 4, and 5 without repetition, how many distinct five-digit "wave numbers" can be formed such that the tens digit and the thousands digit are each larger than their adjacent digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Find the linear operator L such that \( Lu = 0 \) for the function \( u = 2t + e^{-t} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{D^3 + D^2}
Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute \[ \frac{1}{255}\sum_{0\leq n<16}2^n(-1)^{s(n)}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
In the alternating group $A_5$, calculate the probability that a randomly selected pair of distinct elements generates a cyclic subgroup.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{118}}
Determine the cardinality of the set of all functions from the set of natural numbers \( \mathbb{N} \) to the set of real numbers \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Five people are seated around a circular table. Let f represent the number of people sitting next to at least one female and m represent the number of people sitting next to at least one male, where f and m are non-negative integers. Calculate the number of unique ordered pairs (f, m).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Tom decides to open a theater with a capacity of 500 seats. He knows it will cost $5 per square foot of space and he needs 12 square feet for every seat in his theater.	In addition to the cost of land and construction, he also needs to consider infrastructure costs, like electricity and plumbing, which add an additional 15% to the total cost of land and construction. Moreover, the government imposes a 7% tax on the total cost. Tom also plans to set aside 3% of the total cost for annual maintenance fees. Tom's partner will contribute 40% towards the overall cost. Calculate the total amount Tom will need to spend after considering all expenses and the contribution from his partner.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$22,813.47}
How many integer points (x, y, z) satisfy the equation \(35x + 55y + 77z = 1\) and are contained within a cube of side length 30 centered at the origin (0, 0, 0)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the number of ring isomorphisms from \( \mathbb{Z}^n \) to \( \mathbb{Z}^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
You are playing a game in which you have  $3$  envelopes, each containing a uniformly random amount of money between  $0$  and  $1000$  dollars. (That is, for any real  $0 \leq a < b \leq 1000$ , the probability that the amount of money in a given envelope is between  $a$  and  $b$  is  $\frac{b-a}{1000}$ .) At any step, you take an envelope and look at its contents. You may choose either to keep the envelope, at which point you finish, or discard it and repeat the process with one less envelope. If you play to optimize your expected winnings, your expected winnings will be  $E$ . What is  $\lfloor E\rfloor,$  the greatest integer less than or equal to  $E$ ?

*Author: Alex Zhu*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{695}