inclusionAI/Ring-lite-rl-data huggingface dataset:
We define the ridiculous numbers recursively as follows:
[list=a]
[*]1 is a ridiculous number.
[*]If $a$ is a ridiculous number, then $\sqrt{a}$ and $1+\sqrt{a}$ are also ridiculous numbers. 
[/list]
A closed interval $I$ is ``boring'' if
[list]
[*]$I$ contains no ridiculous numbers, and
[*]There exists an interval $[b,c]$ containing $I$ for which $b$ and $c$ are both ridiculous numbers.
[/list]
The smallest non-negative $l$ such that there does not exist a boring interval with length $l$ can be represented in the form $\dfrac{a + b\sqrt{c}}{d}$ where $a, b, c, d$ are integers, $\gcd(a, b, d) = 1$, and no integer square greater than 1 divides $c$. What is $a + b + c + d$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A right triangle and a circle are drawn such that the circle is tangent to the legs of the right triangle. The circle cuts the hypotenuse into three segments of lengths 1,24 , and 3 , and the segment of length 24 is a chord of the circle. Compute the area of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192}
Given the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with its left focus $F(c, 0)$ and eccentricity $\frac{\sqrt{3}}{3}$. Point $M$ lies on the ellipse in the first quadrant, and the line segment formed by intersecting line $FM$ with the circle $x^{2} + y^{2} = \frac{b^{2}}{4}$ has a length of $c$. Calculate the slope of the line $FM$, expressing your answer as a simplified fraction or decimal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
If one side of square \(ABCD\) lies on the line \(y = 2x - 17\) and two other vertices lie on the parabola \(y = x^2\), then the minimum area of the square is \(\quad\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
If there are several heaps of stones on the table, it is said that there are $\textit{many}$ stones on the table, if we can find $50$ piles and number them with the numbers from $1$ to $50$ so that the first pile contains at least one stone, the second - at least two stones,..., the $50$-th has at least $50$ stones. Let the table be initially contain $100$ piles of $100$ stones each. Find the largest $n \leq 10 000$ such that after removing any $n$ stones, there will still be $\textit{many}$ stones left on the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5099}
Two positive integers that only have 1 as a common factor are called coprime numbers. For example, 2 and 7 are coprime, as are 3 and 4. In any permutation of 2, 3, 4, 5, 6, 7, where each pair of adjacent numbers are coprime, there are a total of \_\_\_\_\_\_\_\_ different permutations (answer with a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
After receiving his weekly allowance, a student decided to save 10% of his allowance each day. On the first day, he spent 3/5 of his remaining allowance at the arcade. On the second day, he saved 10% of his remaining allowance and then spent one-third of what was left at the toy store. The next day, again after saving 10%, he spent his last $0.60 at the candy store. What is this student's weekly allowance?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$3.09}
Given the set $S=\{1,2,3,4,5,6,7,8\}$, for $x\in S$, if $x+1 \notin S$ and $x-1 \notin S$, then $x$ is a "good element" of $S$. Among all sets composed of $3$ elements from $S$, the number of sets that do not contain any "good elements" is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Compute
$$
\sum_{k=0}^{100}\left\lfloor\frac{2^{100}}{2^{50}+2^{k}}\right\rfloor .
$$
(Here, if \( x \) is a real number, then \( \lfloor x \rfloor \) denotes the largest integer less than or equal to \( x \).)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101 \cdot 2^{49} - 50}
Calculate the result of the expression $826446281 \times 11 \times 11$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100000000001}
Calculate the integral
$$\int_{\mathbb{R}^2}\exp\left({-\frac{(q^2+p^2)}{2}}\right)\exp\left({-\frac{\left(\cosh(s)q+\sinh(s)p\right)^2+\left(\sinh(s)q+\cosh(s)p\right)^2}{2}}\right)dq\;dp$$
where $s\geq 0$ is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{\cosh(s)}\)}
A person can swim in still water at 6 km/h. If the speed of water is 2 km/h, how many hours will the person take to swim back against the current for a certain distance, if it takes them 3.5 hours?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
You roll three fair six-sided dice. Given that the highest number you rolled is $5$, the expected value of the sum of the three dice can be written as $\frac{a}{b}$ in simplest form. Find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{706}
Consider the rational function $Q(x)$ satisfying $x \cdot Q(x+2018) = (x-2018)Q(x)$ for all $x \neq 2018$ and $0$. Given that $Q(1) = 1$, find the value of $Q(2017)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
Five soccer teams compete, with each pair of teams playing one match: the winner gets 3 points, the loser gets 0 points, and in case of a draw each team gets 1 point. After the matches, the total points of each team do not exceed 9 points, and there are exactly two teams that have the same points. Let the scores of the five teams from highest to lowest be $A, B, C, D, E$ (where there are exactly two equal scores). If $\overline{\mathrm{ABCDE}}$ (the concatenation of the scores) is a multiple of 15, how many draws were there in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The integer $k$ is a \textit{good number}, if we can divide a square into $k$ squares. Find how many good numbers not greater than $2006$ there are.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2003}
Determine all pairs $(a, p)$ of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all elements in these pairs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Determine the number of distinct integers among the numbers
$$
\left\lfloor\frac{1^{2}}{2007}\right\rfloor,\left\lfloor\frac{2^{2}}{2007}\right\rfloor, \quad \cdots, \quad \left\lfloor\frac{2007^{2}}{2007}\right\rfloor.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1506}
In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2 + 10$, $10 + 2$, and $3 + 2 + 2 + 5$ are three such ways.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
each of the positive integers a and c is a 4 - digit integer . if each of the digits 0 through 9 appears in one of these 3 integers , what is the maximum possible value of the sum of a and c ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18395}
Suppose $a,$ $b,$ and $c$ are real numbers such that
\[\frac{ac}{a + b} + \frac{ba}{b + c} + \frac{cb}{c + a} = -9\]and
\[\frac{bc}{a + b} + \frac{ca}{b + c} + \frac{ab}{c + a} = 10.\]Compute the value of
\[\frac{b}{a + b} + \frac{c}{b + c} + \frac{a}{c + a}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A large cube is made up of 27 small cubes. A plane is perpendicular to one of the diagonals of this large cube and bisects the diagonal. How many small cubes are intersected by this plane?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
The ratio of investments of three partners p, q, and r is 7 : 5 : 4, and the ratio of their profits is 7 : 10 : 8. If p invested the money for 2 months and q invested the money for t months, find for how much time did r invest the money?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given the function $f(x) = \ln x$, its tangent line equation at $x=x_0$ is $y=kx+b$. Determine the range of values for $k+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, +\infty)}
For any positive integer  $a$ , define  $M(a)$  to be the number of positive integers  $b$  for which  $a+b$  divides  $ab$ . Find all integer(s)  $a$  with  $1\le a\le 2013$  such that  $M(a)$  attains the largest possible value in the range of  $a$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1680}
Given unit vectors $\vec{a}$ and $\vec{b}$ with an acute angle between them, for any $(x, y) \in \{(x, y) \mid | x \vec{a} + y \vec{b} | = 1, xy \geq 0 \}$, it holds that $|x + 2y| \leq \frac{8}{\sqrt{15}}$. Find the minimum possible value of $\vec{a} \cdot \vec{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
How many positive five-digit integers have the product of their five digits equal to $900$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
Find the largest integer $d$ that divides all three numbers $abc$, $bca$, and $cab$, where $a$, $b$, and $c$ are some nonzero and mutually different digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
A circle touches the extensions of two sides $AB$ and $AD$ of square $ABCD$ with a side length of $2-\sqrt{5-\sqrt{5}}$ cm. From point $C$, two tangents are drawn to this circle. Find the radius of the circle, given that the angle between the tangents is $72^{\circ}$ and it is known that $\sin 36^{\circ} = \frac{\sqrt{5-\sqrt{5}}}{2 \sqrt{2}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{ 5- \sqrt (5)}}
On a game show, a contestant is given 5 keys, each of which opens exactly one of 5 identical boxes. The first box contains $100, the second $500, the third $1,000, the fourth $5,000, and the fifth $10,000. The contestant assigns each key to one of the boxes and wins the total amount of money contained in any boxes that are opened by the keys assigned to them. What is the probability that a contestant will win more than $12,000?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/20}
In the diagram, a large circle and a rectangle intersect such that the rectangle halves the circle with its diagonal, and $O$ is the center of the circle. The area of the circle is $100\pi$. The top right corner of the rectangle touches the circle while the other corner is at the center of the circle. Determine the total shaded area formed by the parts of the circle not included in the intersection with the rectangle. Assume the intersection forms a sector.
[Diagram not shown: Assume descriptive adequacy for the composition of the circle and the rectangle.]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\pi}
Let $a$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y)$ satisfy all of the following conditions:
\begin{enumerate}
    \item $\frac{a}{2} \le x \le 2a$
    \item $\frac{a}{2} \le y \le 2a$
    \item $x+y \ge a$
    \item $x+a \ge y$
    \item $y+a \ge x$
\end{enumerate}
The boundary of set $S$ is a polygon with
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In three of the six circles of a diagram, the numbers 4, 14, and 6 are written. In how many ways can natural numbers be placed in the remaining three circles so that the product of the triplets of numbers along each of the three sides of the triangular diagram are the same?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is the smallest value of \( n \) needed for the \( n \)th order Taylor polynomial \( T_n(x,0) \) of \( \cos x \) centered at \( 0 \) to guarantee an error \( |\cos x - T_n(x,0)| \) less than \( 10^{-3} \) for \( x \) in the interval \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Tessa has a figure created by adding a semicircle of radius 1 on each side of an equilateral triangle with side length 2, with the semicircles oriented outwards. She then marks two points on the boundary of the figure. What is the greatest possible distance between the two points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
For the parabola $C$, suppose a line $l$ passes through the focus $F$ of $C$, and the angle between $l$ and the axis of symmetry of $C$ is $\frac{\pi}{4}$. If the chord intercepted by $l$ on $C$ has a length of 4, then the distance from the focus of $C$ to its vertex is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
解方程$2 \sqrt { x ^ { 2 } - 1 2 1 } + 1 1 \sqrt { x ^ { 2 } - 4 } = 7 \sqrt { 3 } x .$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
The sum of the first $n$ terms of an infinite sequence $\{a_n\}$ is $S_n$. If for any positive integer $n$, $S_n$ belongs to $\{k_1,k_2,k_3,…,k_{10}\}$, then the maximum number of possible values for $a_{10}$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Find the limit of the following expression as x approaches 0 from the right:

$$\lim_{x \to 0+}\frac{x^{(\sin x)^x}-(\sin x)^{x^{\sin x}}}{x^3}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$$\frac{1}{6}$$}
Triangle $ABC$ has $AB = AC$. Point $D$ is on side $\overline{BC}$ so that $AD = CD$ and $\angle BAD = 36^\circ$. Find the degree measure of $\angle BAC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
A circle is circumscribed around the right triangle $ABC$ ($\widehat{C} = 90^\circ$). Let $CD$ be the altitude of the triangle. A circle with center at $D$ passes through the midpoint of the arc $\widetilde{AB}$ and intersects $AB$ at point $M$. Find $|CM|$, if $|AB|=c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{c\sqrt{2}}{2}}
We have an $n$-gon, and each of its vertices is labeled with a number from the set $\{1, \ldots, 10\}$. We know that for any pair of distinct numbers from this set there is at least one side of the polygon whose endpoints have these two numbers. Find the smallest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Question 78: Given $x, y, z > 0$, find the minimum value of $2 \sqrt{(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)}-\sqrt{\left(1+\frac{x}{y}\right)\left(1+\frac{y}{z}\right)}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}+1}
Two 5-digit positive integers are formed using each of the digits from 0 through 9 once. What is the smallest possible positive difference between the two integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{247}
Evaluate the limit:

$$\lim\limits_{n\to\infty}{\sum\limits_{k=n}^{5n} \frac{n}{k}{k \choose n}(\frac{1}{5})^{n}(\frac{4}{5})^{k-n}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
$x_{1}, x_{2}, x_{3}, x_{4}$ 为互不相等的正实数， $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 为 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意顺序排列, $X=\max \left\{\min \left\{x_{i 1}, x_{i 2}\right\}, \min \left\{x_{i 3}, x_{i 4}\right\}, Y=\min \left\{\max \left\{x_{i 1}, x_{i 2}\right\}, \max \left\{x_{i 3}, x_{i 4}\right\}\right.\right.$, 求 x 大于 y 的概率.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{4}
How many positive integers less than 2005 are relatively prime to 1001?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1442}
There are 3 positions to be filled by selecting from 4 college graduates. If each position must select at least 1 person (not all 4 college graduates necessarily get selected), then the number of different ways to select is \_\_\_\_\_\_\_\_ (answer with a specific number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Two trains of some length and 200 m are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. They will be clear of each other from the moment they meet in 14.998800095992321 seconds. What is the length of the first train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99.98}
4. In the stands of the hockey arena, there are several rows with 168 seats in each row. For the final match, 2016 students from several sports schools were invited as spectators, with no more than 40 from each school. Students from any school must be seated in one row. What is the minimum number of rows that must be in the arena to ensure this can always be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
For each integer $1 \le j \le 2017$, let $S_j$ denote the set of integers $0 \le i \le 2^{2017} - 1$ such that $\left\lfloor \frac{i}{2^{j-1}} \right\rfloor$ is an odd integer. Let $P$ be a polynomial such that
\[P\left(x_0, x_1, \ldots, x_{2^{2017} - 1}\right) = \prod_{1 \le j \le 2017} \left(1 - \prod_{i \in S_j} x_i\right).\]
Compute the remainder when
\[ \sum_{\left(x_0, \ldots, x_{2^{2017} - 1}\right) \in \{0, 1\}^{2^{2017}}} P\left(x_0, \ldots, x_{2^{2017} - 1}\right)\]
is divided by $2017$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1840}
Let $A > 0$ be a positive real number and $n \geq 4$ be an integer. Evaluate the integral:

$$ \int_0^\infty \log \left (1+\left (\frac A {\sqrt t}\right)^n\right) \, dt $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\pi A^2 \csc\left(\frac{2\pi}{n}\right)\)}
At 15:00 there were 20 students in the computer lab. At 15:03 and every three minutes after that, 4 students entered the lab. If at 15:10 and every ten minutes after that 8 students left the lab, how many students were in the computer lab at 15:44?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Juan makes a list of $2018$ numbers. The first number is $1$. Each subsequent number is obtained by adding one of the numbers $1, 2, 3, 4, 5, 6, 7, 8,$ or $9$ to the previous number. Given that none of the numbers in the list ends in $0$, what is the largest possible value for the last number on the list?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17929}
Given the function y = f(x) = x + sqrt(13 - 2ax), where a is a positive integer, find the maximum value of y such that it is a positive integer. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given the equation \( 4(\sqrt{x} + \sqrt{y-1} + \sqrt{z-2}) = x + y + z + 9 \), calculate the value of \( xyz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Given real numbers \( x \) and \( y \in (1, +\infty) \), and \( x y - 2 x - y + 1 = 0 \). Find the minimum value of \( \frac{3}{2} x^{2} + y^{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
19. (3 points) A male and a female track and field athletes are practicing running on a 110-meter slope (the top of the slope is $A$, the bottom is $B$). Both start from point $A$ at the same time, running back and forth between $A$ and $B$ continuously. If the male athlete's uphill speed is 3 meters per second and downhill speed is 5 meters per second; the female athlete's uphill speed is 2 meters per second and downhill speed is 3 meters per second, then the location of their second face-to-face meeting is $\qquad$ meters away from point $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47\frac{1}{7}}
John has a donkey, an elephant, a giraffe, and a hippopotamus. The elephant weighs 3 tons (a ton is 2000 pounds). The donkey weighs 90% less than the elephant. The giraffe weighs 1.5 tons and the hippopotamus weighs 4000 kg. Additionally, John bought a bag of food for each of them which weighs 5 lbs, 16 oz, 3 kg, and 5000 g, respectively. What is the combined weight of the animals and their bags of food in pounds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18442}
The geometric sequence $\{a_n\}$ satisfies: $a_1=a$ ($a>0$), and $a_{1}+1$, $a_{2}+2$, $a_{3}+3$ form a geometric sequence. If $\{a_n\}$ is unique, then the value of $a$ equals \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {1}{3}}
Thomas started saving for a car almost 2 years ago. His weekly allowance was some amount for the first year. In the second year, he got a job that pays $9 an hour at a coffee shop and worked 30 hours a week, so his parents discontinued his allowance. If the car he wants to buy is $15,000 and he spends $35 a week on himself, Thomas needs $2000 more to buy the car by the end of the 2 years. What was Thomas's weekly allowance in the first year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$15}
Given the set M = {1,2,3,4,5,6,7} and the proposition p: ∀n∈M, n > 1, what is the negation of proposition p? Express your answer as a logical statement in the form of ∃n∈M, [condition].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{∃n∈M, n ≤ 1}
For a positive integer $ n$, let $ \theta(n)$ denote the number of integers $ 0 \leq x < 2010$ such that $ x^2 \minus{} n$ is divisible by $ 2010$. Determine the remainder when $ \displaystyle \sum_{n \equal{} 0}^{2009} n \cdot \theta(n)$ is divided by $ 2010$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In each cell of a \(75 \times 75\) table, a natural number is written. Each row contains at least 15 different numbers, and in every three consecutive rows, there are no more than 25 different numbers. What is the maximum number of different numbers that the table can contain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{385}
A trapezoidal room has dimensions of 25 feet (length), 15 feet (width), 12 feet (height1) and 8 feet (height2). What is the cost of white washing the four walls of the room at Rs. 10 per square feet, if there are two doors with dimensions of 6 feet * 3 feet and three windows of dimensions 4 feet * 3 feet each, and considering an irregular ceiling decoration that subtracts 10 square feet from the total area to be painted?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7580}
Given that the Earth's global mean temperature follows a normal distribution $N(\mu, \sigma)$, and we have $N$ samples, denoted as $T_1, \cdots, T_N$. Let $T_{maxN}$ be the maximum of the first $N$ samples. What is the probability of drawing three consecutive samples, $T_{N+1}, T_{N+2}, T_{N+3}$, such that each subsequent sample is greater than the previous and the maximum of the first $N$ samples, i.e., $P(T_{N+3} > T_{N+2} > T_{N+1} > T_{maxN})$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{(N+1)(N+2)(N+3)}\)}