inclusionAI/Ring-lite-rl-data huggingface dataset:
The number of non-empty proper subsets of the set $\{y \in \mathbb{N} | y = -x^2 + 6, x \in \mathbb{N}\}$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
"That" and "this," and half of "that" and "this" - how many percent is this of three-quarters of "that" and "this"?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200\%}
What will be the difference between simple and compound interest at 8% per annum on a certain sum after 4 years if the difference is 40.48896000000036?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
已知函数 y=f(x) 的图像既关于点 (1,1) 中心对称，又关于直线 x+y=0 轴对称. 若 x\in(0,1) 时，f(x)=\log_2(x+1)，则 f(\log_210) 的值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
What is the absolute value of the difference between the squares of 101 and 99?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
In $\triangle ABC$, $\angle A = 60^{\circ}$ and $AB > AC$. Point $O$ is the circumcenter, and the altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ lie on segments $BH$ and $HF$, respectively, such that $BM = CN$. Find the value of $\frac{MH + NH}{OH}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Given the infinite series representations:
$$a = 1 + \frac{x^3}{3!} + \frac{x^6}{6!} + \cdots$$
$$b = \frac{x}{1!} + \frac{x^4}{4!} + \frac{x^7}{7!} + \cdots$$
$$c = \frac{x^2}{2!} + \frac{x^5}{5!} + \frac{x^8}{8!} + \cdots$$
Let $u = a^2 + b^2 + c^2$. Determine the value of $u(\ln 2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{3}}
设U={1,2,3,⋯,n}(n≥3)，S是U的一个子集.如果不属于S的一个元素出现在U的一个排列P的某处，并且在该排列P中它处在S中的两个元素之间，那么我们称S是由排列P分裂出来的（例如排列13542可分裂出子集｛1,2,3｝，而不能分裂出子集｛3,4,5｝），试求最大正整数m，使得对U的任意m个子集（每个子集至少有2个元，至多有n-1个元），都存在U的一个排列分裂出所有这m个子集.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n-2}
Find the minimum possible value of $\sqrt{58-42 x}+\sqrt{149-140 \sqrt{1-x^{2}}}$ where $-1 \leq x \leq 1$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{109}}
In a certain month of a certain year, there are exactly 5 Fridays. Furthermore, the first and last day of this month are not Fridays. What day of the week is the 12th day of the month?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Monday}}
Find the smallest positive integer  $n$  such that if we color in red  $n$  arbitrary vertices of the cube , there will be a vertex of the cube which has the three vertices adjacent to it colored in red.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Every cell of a $3 \times 3$ board is colored either red or blue. Find the number of all possible colorings such that there are no $2 \times 2$ squares where all cells are red.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{417}
Let \( T \) be a right triangle with sides having lengths 3, 4, and 5. A point \( P \) is called awesome if \( P \) is the center of a parallelogram whose vertices all lie on the boundary of \( T \). What is the area of the set of awesome points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/2}
Add one pair of brackets to the expression:

\[1 + 2 \times 3 + 4 \times 5 + 6\]

so that the resulting expression has a valid mathematical value. For example, \(1 + 2 \times (3 + 4 \times 5) + 6 = 53\). What is the largest possible value that one can make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77}
8. In a certain World Cup, the Brazilian team has a probability of $\frac{1}{2}$ to win against each opponent, a probability of $\frac{1}{3}$ to draw, and a probability of $\frac{1}{6}$ to lose. Winning a match earns 3 points, drawing earns 1 point, and losing earns 0 points. It is known that each team plays three matches in the group stage, and 4 points or more are required to advance from the group. In the knockout stage, if the teams draw, they go to a penalty shootout. It is known that Brazil's probability of winning a penalty shootout is $\frac{3}{5}$. What is the probability that Brazil will win the championship and have exactly one penalty shootout in the four knockout matches? $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{12}}
Let  $S$  be a subset of  $\{0,1,2,\dots ,9\}$ . Suppose there is a positive integer  $N$  such that for any integer  $n>N$ , one can find positive integers  $a,b$  so that  $n=a+b$  and all the digits in the decimal representations of  $a,b$  (expressed without leading zeros) are in  $S$ . Find the smallest possible value of  $|S|$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given that the eccentricity of a hyperbola is $2$, what is the acute angle formed by its two asymptotes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Calculate the number of subgraphs of a graph with two disconnected components, one with 4 vertices and 4 edges, and the other with 2 vertices and 1 edge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Let $a, b, c, d$ be natural numbers such that $a + b + c + d = 2018$. Find the minimum value of the expression:
$$E = (a-b)^2 + 2(a-c)^2 + 3(a-d)^2 + 4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let $p(x)=2+4x+3x^2+5x^3+3x^4+4x^5+2x^6$. For $k$ with $0<k<5$, deline
$$I_k=\int_{0}^{\infty}\frac{x^k}{p(x)}dx$$.
For which $k$ is $I_k$ smallest.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The area of rhombus \(ABCD\) is 2. The triangle \(ABD\) has an inscribed circle that touches the side \(AB\) at point \(K\). A line \(KL\) is drawn through point \(K\) parallel to the diagonal \(AC\) of the rhombus (the point \(L\) lies on the side \(BC\)). Find the angle \(BAD\) given that the area of triangle \(KLB\) is \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \arcsin \sqrt{a}}
Mary just arrived at the beach. She has 4 times as many towels as Frances does. The total weight of their towels is 60 pounds. If Frances's towels weigh 128 ounces, how many towels does Mary have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $N$ be the largest positive integer that can be expressed as a 2013-digit base -4 number. What is the remainder when $N$ is divided by 210?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Determine the Jordan Normal Form of a nilpotent matrix $N \in \text{Mat}(10 \times 10, \mathbb{C})$ given that $\dim \ker N = 3$, $\dim \ker N^2 = 6$, and $\dim \ker N^3 = 7$. Provide the sizes of the Jordan blocks in the form of a list.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[6, 2, 2]}
Determine the number of subgroups in an abelian group of order 1,000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Evaluate the limit: $$\lim_{x \to 0}\left(\Gamma\left(\frac{x}{3}\right) - \frac{3}{x} + 3\gamma \right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\gamma}
Evaluate the limit: $$\lim_{n\rightarrow\infty}\sum_{k=2}^n\frac{1-e^{-(x_k-x_{k-1})}}{1+e^{-(x_k-x_{k-1})}}$$ where the points $\{x_1,..,x_n\} \in [0,1]$ are ordered $x_1 < x_2 < ... < x_n$ and form a partition of $[0,1]$. Moreover, the maximum difference $x_j - x_{j-1}$ tends to zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine all weak limit points of the sequence of measures \( \{\mu_n\} \) on \((\mathbb{R}, B)\), where \( \mu_n = U_{\{\frac{1}{n}, \frac{2}{n}, \dots, \frac{n-1}{n}, 1\}} \) and \( U \) denotes the uniform distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{U_{[0,1]}}
Twenty circles are drawn in the plane, all passing through the origin, with no two circles tangent at the origin. Additionally, except for the origin, no three circles pass through a common point. Determine the number of regions created in the plane by these circles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{211}
Find the least number which when divided by 38 and another number leaves a remainder of 1 in each case. The least number is 115. What is the other divisor?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{57}
Four lighthouses are located at points $A$, $B$, $C$, and $D$. The lighthouse at $A$ is $5$ kilometers from the lighthouse at $B$, the lighthouse at $B$ is $12$ kilometers from the lighthouse at $C$, and the lighthouse at $A$ is $13$ kilometers from the lighthouse at $C$. To an observer at $A$, the angle determined by the lights at $B$ and $D$ and the angle determined by the lights at $C$ and $D$ are equal. To an observer at $C$, the angle determined by the lights at $A$ and $B$ and the angle determined by the lights at $D$ and $B$ are equal. The number of kilometers from $A$ to $D$ is given by $\frac {p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are relatively prime positive integers, and $r$ is not divisible by the square of any prime. Find $p$ + $q$ + $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Find the limit of the integral $$\lim_{n \to \infty} \int_0^n \frac{dx}{1+n^2\cos^2x}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the kernel of the homomorphism \( k: R[x,y] \to R[x,y] \) defined by \( f(x,y) \mapsto \frac{\partial^2 f}{\partial x \partial y} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}[x] + \mathbb{R}[y]}
How can 50 cities be connected with the smallest number of airline routes so that it is possible to travel from any city to any other city with no more than two layovers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Determine the optimal reserve price for a second-price sealed-bid auction where a seller offers a single private value good to two potential buyers. The buyers' valuations are uniformly distributed between 0 and 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110.8333333}
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2027\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution. Find the minimum possible value of \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1014}
Three cards are dealt at random from a standard deck of 52 cards.  What is the probability that the first card is a 4, the second card is a $\clubsuit$, and the third card is a 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{663}}
Find the sum for positive integers $n$ and $m$:
$$S(n, m) = \sum_{i=0}^n \frac{1}{2^{m+i+1}}\binom{m+i}{i} + \sum_{i=0}^m \frac{1}{2^{n+i+1}}\binom{n+i}{i}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compute the number of positive integers $n$ between $2017$ and $2017^2$ such that $n^n \equiv 1 \pmod{2017}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30576}
Given $2^{a}=3$ and $\log_{3}5=b$, then $\log_{15}20=$ \_\_\_\_\_\_ (express in terms of $a$ and $b$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2+ab}{a+ab}}
Let $ABCD$ be a convex quadrilateral with $AC \perp BD$, and let $P$ be the intersection of $AC$ and $BD$. Suppose that the distance from $P$ to $AB$ is $99$, the distance from $P$ to $BC$ is $63$, and the distance from $P$ to $CD$ is $77$. What is the distance from $P$ to $AD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{231}
Let the probability of germination for each seed be 0.9, and 1000 seeds have been planted. For each seed that does not germinate, 2 more seeds need to be replanted. Let X be the number of replanted seeds. Calculate the expected value of X.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Jill and her brother Jack are going apple picking. Jill's basket can hold twice as much as Jack's basket when both are full. Jack's basket is full when it has 12 apples and currently has space for 4 more. Each apple has an average weight of 150 grams. The empty baskets themselves also have weight; Jill's basket weighs 300 grams and Jack's basket weighs 200 grams. What is the total weight of apples in Jill's basket when it is full?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3600}
A standard six-sided die is rolled repeatedly until a 6 appears. Calculate the probability that the sum of all rolls, including the roll that results in a 6, is even.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{7}}
What percent of the first $20$ positive integers are divisible by $3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30%}
In a day, the hands of a clock are at right angles a certain number of times. How many times are they at right angles in a day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
A fair coin is flipped 30 times. Let \( X \) denote the number of heads among the first 20 coin flips, and \( Y \) denote the number of heads among the last 20 coin flips. Compute the correlation coefficient of \( X \) and \( Y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Consider a 4 × 4 grid of squares, where each square is colored either black or white. Two colorings are considered identical if one can be transformed into the other by any combination of flipping, rotating, or swapping the two colors (changing all black squares to white and vice versa). Determine the number of distinct colorings of the grid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4324}
Given that a group of students is sitting evenly spaced around a circular table and a bag containing 120 pieces of candy is circulated among them, determine the possible number of students if Sam picks both the first and a final piece after the bag has completed exactly two full rounds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Let $ \mathcal{C}$ be the hyperbola $ y^2 \minus{} x^2 \equal{} 1$.  Given a point $ P_0$ on the $ x$-axis, we construct a sequence of points $ (P_n)$ on the $ x$-axis in the following manner: let $ \ell_n$ be the line with slope $ 1$ passing passing through $ P_n$, then $ P_{n\plus{}1}$ is the orthogonal projection of the point of intersection of $ \ell_n$ and $ \mathcal C$ onto the $ x$-axis. (If $ P_n \equal{} 0$, then the sequence simply terminates.)

Let $ N$ be the number of starting positions $ P_0$ on the $ x$-axis such that $ P_0 \equal{} P_{2008}$. Determine the remainder of $ N$ when divided by $ 2008$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{254}
Fill the numbers $1$ to $9$ into a $3 \times 3$ grid, such that in each row, the number to the right is greater than the number to the left, and in each column, the number below is greater than the number above. Given that $1$, $4$, and $9$ are already placed in the grid, determine the number of different ways to fill the remaining 6 integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Each triangle in this figure is an isosceles right triangle. The length of $\overline{BC}$ is 2 units. What is the number of units in the perimeter of quadrilateral $ABCD$? Express your answer in simplest radical form.

[asy]
draw((0,0)--(10,0)--(0,10)--cycle);
draw((10,0)--(0,10)--(10,20)--cycle);
draw((1,0)--(1,1)--(0,1));
draw((1,11)--(2,10)--(1,9));
label("$A$",(0,0),S);
label("$B$",(10,0),S);
label("$C$",(10,20),N);
label("$D$",(0,10),W);

[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4+\sqrt{2}}
Consider two distinct positive integers $a$ and $b$ that have integer arithmetic, geometric, and harmonic means. Find the minimal value of $|a-b|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Consider $m$ segments on the real line. Each segment has its two endpoints in the set of integers $\{1, 2, \ldots, 2024\}$, and no two segments have the same length. No segment is entirely contained in another segment, but two segments may partially overlap each other.

What is the maximum value of $m$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
Find the distance from the ellipsoid \(x^2+y^2+4z^2=4\) to the plane \(x+y+z=6\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
已知正实数 $x, y$ 满足: $\frac{1}{x+3 y}+\frac{1}{2 x+y}=1$, 则 $x+y$ 的最小值是$\frac{a}{b}(c+d \sqrt{d})$，求$a+b+c+d$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Calculate the limit: $$\lim_{{n \rightarrow \infty}}{\prod_{j=1}^{n}\Big(1+\frac{2j-1}{2n}\Big)^{\frac{1}{2n}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{\sqrt{e}}}
Find the largest number \( m \) such that for any positive numbers \( a, b, \) and \( c \) whose sum equals 1, the following inequality holds:

$$
\sqrt{\frac{a b}{c+a b}}+\sqrt{\frac{b c}{a+b c}}+\sqrt{\frac{c a}{b+c a}} \geqslant m
$$

(E. Emelyanov)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the ratio of the legs in a right triangle, if the triangle formed by its altitudes as sides is also a right triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{-1 + \sqrt{5}}{2}}}
What is the maximum possible order of a group $G$ in which every element $g$ satisfies $g^3 = e$, the identity element?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Let \( OP \) be the diameter of the circle \( \Omega \), and let \( \omega \) be a circle with its center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). The chord \( OB \) of circle \( \Omega \) intersects the second circle at point \( A \). Find the length of the segment \( AB \) if \( BD \cdot BC = 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
Let $a, b, c, x, y,$ and $z$ be complex numbers such that \[ a = \frac{b+c}{x-2}, \ b = \frac{c+a}{y-2}, \ c = \frac{a+b}{z-2}. \] If $xy + yz + xz = 67$ and $x + y + z = 2010$, find the value of $xyz$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5892}
For $\pi \le \theta < 2\pi$, let
\begin{align*} P &= \frac12\cos\theta - \frac14\sin 2\theta - \frac18\cos 3\theta + \frac{1}{16}\sin 4\theta + \frac{1}{32} \cos 5\theta - \frac{1}{64} \sin 6\theta - \frac{1}{128} \cos 7\theta + \cdots \end{align*}
 and
\begin{align*} Q &= 1 - \frac12\sin\theta -\frac14\cos 2\theta + \frac18 \sin 3\theta + \frac{1}{16}\cos 4\theta - \frac{1}{32}\sin 5\theta - \frac{1}{64}\cos 6\theta +\frac{1}{128}\sin 7\theta + \cdots \end{align*}
 so that $\frac{P}{Q} = \frac{2\sqrt2}{7}$. Then $\sin\theta = -\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Let \( N \) denote the number of subsets of \(\{1,2,3, \ldots, 100\}\) that contain more prime numbers than multiples of 4. Compute the largest integer \( k \) such that \( 2^{k} \) divides \( N \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
Given that a^2 - b^2 = 10 and a * b = some value, find the value if a^4 + b^4 = 228.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(8\)}
A motorist embarks on a complex journey lasting 8 hours. He begins by driving for 2 hours at a speed of 60 kmph, followed by 3 hours of travel on a train at a constant speed of 100 kmph. He then takes a 45-minute break before driving for the remaining portion of the trip at varying speeds: for the first 75 minutes, he drives at 72 kmph, then slows down to 50 kmph for 95 minutes. Calculate the total distance covered by the motorist throughout the journey.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{589.17}
Let's find such an $n$ and such numbers $a_{i}, b_{i}, c_{i},(i=1,2, \ldots, n)$, for which the points $(x, y)$ satisfying the condition

$$
\sum_{i=1}^{n}\left|a_{i} x+b_{i} y+c_{i}\right|=10
$$

when plotted on the coordinate system, form the vertices of a regular octagon!
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $T: \mathcal P_2(\Bbb R) \rightarrow \mathcal P_2(\Bbb R)$ be a linear transformation defined by $T(p(x)) = p(1)(x^2 + x) + p(0)$. Consider the standard ordered basis $\alpha = \{x^2, x, 1\}$ for $\mathcal P_2(\Bbb R)$. Determine the matrix $[T]_\alpha$ that represents $T$ with respect to basis $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\begin{bmatrix}1&1&1\\1&1&1\\0&0&1\end{bmatrix}}
a 180 meter long train crosses a man standing on the platform in 9 sec . what is the speed of the train ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given \( x \in \mathbf{R} \), the sum of the maximum and minimum values of the function \( f(x)=\max \left\{\sin x, \cos x, \frac{\sin x+\cos x}{\sqrt{2}}\right\} \) is equal to?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \frac{\sqrt{2}}{2}}
The function \( f(x) = -\log_{\frac{1}{2}}\left(x^{2} - ax - a\right) \) is a decreasing function on the interval \( (-\infty, 1 - \sqrt{3}) \). Find the range of values for \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(1 - \sqrt{3}) \leq a \leq 2}
Let  $S$  be the set of positive integers not divisible by  $p^4$  for all primes  $p$ . Anastasia and Bananastasia play a game.

At the beginning, Anastasia writes down the positive integer  $N$  on the board. Then the players take moves in turn; Bananastasia moves first. On any move of his, Bananastasia replaces the number  $n$  on the blackboard with a number of the form  $n-a$ , where  $a\in S$  is a positive integer. On any move of hers, Anastasia replaces the number  $n$  on the blackboard with a number of the form  $n^k$ , where  $k$  is a positive integer. Bananastasia wins if the number on the board becomes zero.

Compute the second-smallest possible value of  $N$  for which Anastasia can prevent Bananastasia from winning.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{625}