inclusionAI/Ring-lite-rl-data huggingface dataset:
A $2008 \times 2009$ rectangle is divided into unit squares. In how many ways can you remove a pair of squares such that the remainder can be covered with $1 \times 2$ dominoes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4068434225296}
Let \(\mathcal{P}\) be a parabola, and let \(V_{1}\) and \(F_{1}\) be its vertex and focus, respectively. Let \(A\) and \(B\) be points on \(\mathcal{P}\) so that \(\angle A V_{1} B=90^{\circ}\). Let \(\mathcal{Q}\) be the locus of the midpoint of \(AB\). It turns out that \(\mathcal{Q}\) is also a parabola, and let \(V_{2}\) and \(F_{2}\) denote its vertex and focus, respectively. Determine the ratio \(F_{1} F_{2} / V_{1} V_{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7/8}
Two people start at the same time and move with the same pace on a 5 x 5 square grid, one starting from the bottom left corner and the other from the top right corner. Calculate the probability that they meet on the grid. Express your answer as a fraction in the form \( \frac{a}{b} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{70}}
A sequence is defined as \( {a}_{1} = {2025} \) and for all \( n \geq  2 \) ,
 \( {a}_{n} = \frac{{a}_{n - 1} + 1}{n} \)
 Determine the smallest \( k \) such that \( {a}_{k} < \frac{1}{2025} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2027
}
Find the distribution of the random variable \( Y = X^2 \), where \( X \sim N(0,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\chi^2(1)}
Evaluate the expression $(1^1+2^1+2^{2}+\dots)(1^1+3^1+3^{2}+\dots)(1^1+5^1+5^{2}+\dots)\dots$ using the properties of the Riemann zeta function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{12}}
6 animals of a circus have to be placed inside some cages, one in each cage. If a certain number of cages are too small for 6 of the animals, there are 240 ways of caging the animals. How many cages are too small for the animals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A line divides the length of an arc of a circle in the ratio 1:3. In what ratio does it divide the area of the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi - 2}{3\pi + 2}}
Find all positive integers $k$ such that there exist positive integers $a$ and $b$ satisfying the equation:
\[
a^2 + 4 = (k^2 - 4)b^2.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Form a four-digit number without repeating digits using the numbers 0, 1, 2, 3, 4, 5, 6, where the sum of the digits in the units, tens, and hundreds places is even. How many such four-digit numbers are there? (Answer with a number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{324}
Determine the number of terms needed in the series expansion of \( \int_{0}^{1} \frac{e^x-1}{x} \, dx \) to approximate the integral with an error less than or equal to \( 10^{-6} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the area enclosed by the curve $g(x)$, the lines $x=-3$ and $x=5$, and the $x$-axis, where $g(x)$ is the inverse of $f(x)=x^3+3x+1$. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{2}}
Compute the largest possible number of distinct real solutions for \( x \) to the equation
 \( {x}^{6} + a{x}^{5} + {60}{x}^{4} - {159}{x}^{3} + {240}{x}^{2} + {bx} + c = 0, \)
 where \( a,b \) ,and \( c \) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4
}
Given \( x \in (0,1) \) and \( \frac{1}{x} \notin \mathbf{Z} \), define
\[ 
a_{n}=\frac{n x}{(1-x)(1-2 x) \cdots(1-n x)} \quad \text{for} \quad n=1,2, \cdots 
\]

A number \( x \) is called a "good number" if and only if it makes the sequence \(\{a_{n}\}\) satisfy 
\[ 
a_{1}+a_{2}+\cdots+a_{10} > -1 \quad \text{and} \quad a_{1} a_{2} \cdots a_{10} > 0 
\]

Find the sum of the lengths of all intervals on the number line that represent all such "good numbers".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61/210}
Let \( X_{1}, X_{2}, \ldots, X_{100} \) be a sequence of mutually distinct nonempty subsets of a set \( S \). Any two sets \( X_{i} \) and \( X_{i+1} \) are disjoint and their union is not the whole set \( S \), that is, \( X_{i} \cap X_{i+1}=\emptyset \) and \( X_{i} \cup X_{i+1} \neq S \), for all \( i \in \{1, \ldots, 99\} \). Find the smallest possible number of elements in \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A spaceship travels randomly between three planets: A, B, and C. The journey starts at planet A and ends when the spaceship reaches planet C. The transition probabilities and travel times (in months) between the planets are given in the matrices below. Calculate the expected time (in months) for the spaceship to reach planet C starting from planet A.

**Probability Matrix**

| Origin | Destination A | Destination B | Destination C |
|--------|---------------|---------------|---------------|
| A      | 0.05          | 0.90          | 0.05          |
| B      | 0.50          | 0.50          | 0.00          |
| C      | 0.00          | 0.00          | 1.00          |

**Travel Time Matrix (in months)**

| Origin | Destination A | Destination B | Destination C |
|--------|---------------|---------------|---------------|
| A      | 5             | 2             | 10            |
| B      | 3             | 3             | 0             |
| C      | 0             | 0             | 0             |
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{159}
The sequence $ (a_n)$ is given by $ a_1=1,a_2=0$ and:

$ a_{2k+1}=a_k+a_{k+1}, a_{2k+2}=2a_{k+1}$ for $ k \in \mathbb{N}.$

Find $ a_m$ for $ m=2^{19}+91.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
91
}
Determine the maximum value of \(|f'''(x)|\) for a function \(f\) that is infinitely differentiable on the real numbers, given that \(f(0) = 1\), \(f'(0) = 1\), \(f''(0) = 2\), and \(f(1) < 5\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find an element of order 5 in the field \( \mathbb{Z}_3[x]/\langle x^4 + x + 2 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2x^3 + x + 2}
Calculate the divergence \( \nabla \cdot \hat{n} \) where \( \hat{n} \) is a unit vector normal to a cylinder of radius \( R \) and infinite length \( L = \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{R}}
Given the function $f(x)={x}^{3}+a{x}^{2}+bx+{a}^{2},$, which has an extreme value of $10$ at $x=1$, find the value of $a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
The diagonal of an isosceles trapezoid is $a$, and the midline is $b$. Find the height of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{a^2 - b^2}}
Unit circle \(\Omega\) has points \(X, Y, Z\) on its circumference so that \(X Y Z\) is an equilateral triangle. Let \(W\) be a point other than \(X\) in the plane such that triangle \(W Y Z\) is also equilateral. Determine the area of the region inside triangle \(W Y Z\) that lies outside circle \(\Omega\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{3} - \pi}{3}}
Find the expression for the supremum of the absolute values of the inner products between a unit vector $x$ in a Hilbert space $H$ and vectors $y$ in a subspace $S$ with norm less than or equal to 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\| P_S x \|}
Max likes to collect model trains. He asks for one for every birthday of his, and asks for two each Christmas. Max always gets the gifts he asks for, and asks for these same gifts every year for 5 years. At the end of the 5 years, his parents give him a certain multiple of the number of trains he already has. Max has 45 trains now. What is the ratio of the number of trains his parents gave him at the end of the 5 years to the number of trains he already had?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third monkey takes the remaining bananas from the pile, keeps one-twelfth of them, and divides the rest equally between the other two. Given that each monkey receives a whole number of bananas whenever the bananas are divided, and the numbers of bananas the first, second, and third monkeys have at the end of the process are in the ratio $3: 2: 1$, what is the least possible total for the number of bananas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{408}
An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, calculate the ratio of the shorter side to the longer side of this frame. Express your answer as a simplified ratio in the form a:b.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:3}
Given a monotonic function f(x) defined on (0, +∞), for ∀x ∈ (0, +∞), it holds that f[f(x)-log2(x)] = 3. Find the interval in which the solution of the equation f(x)-f''(x)=2 lies. Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,2)}
Determine the outer normal of the set of $k \times k$ positive definite symmetric matrices, $S^k_+$, as a subset of the set of $k \times k$ symmetric matrices, $S^k$, with respect to the entry-wise inner product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-S^k_+}
Let \( A = \{1, 2, 3, 4, 5, 6\} \). Determine the number of distinct bijective functions \( f: A \rightarrow A \) such that \( f(f(f(n))) = n \) for all \( n \in A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
Suppose $A_1, A_2, A_3, \ldots, A_{20}$ is a 20-sided regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon, but the sides are not the sides of the polygon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{640}
The hyperbola $C$ has its center at the origin and its foci on the $x$-axis. $C$ intersects the directrix of the parabola $y^{2}=16x$ at points $A$ and $B$, with $|AB|=4 \sqrt {3}$. Determine the length of the real axis of $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In a game where each pancake is placed randomly on one of N stacks, calculate the expected number of additional pancakes needed to cover the (k+1)-th pancake on a stack after the k-th pancake has been covered.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N}
In a queue of 100 people waiting to enter a hall with 100 numbered seats from 1 to 100, the first person chooses any seat and sits down. Each subsequent person, from the 2nd to the 100th, sits in their corresponding seat number if it is available; otherwise, they choose any unoccupied seat. How many ways can the 100 seats be occupied such that the 100th person sits in seat number 100?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{98}}
In a division sum, the divisor is 10 times the quotient and some times the remainder. The remainder is 46, and the dividend is 5290. What is the ratio of the divisor to the remainder?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5:1}
Calculate the volume of the solid generated by revolving the curve defined by the parametric equations \(x(t) = a(t - \sin(t))\), \(y(t) = a(1 - \cos(t))\), and \(z(t) = a(1 - \cos(t))\sin(\theta)\) around the x-axis, where \(t\) and \(\theta\) range from \([0, 2\pi]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\pi^2 a^3}
Two lines, parallel to the bases of a trapezoid, divide each of the lateral sides into three equal parts. The entire trapezoid is divided into three sections by these lines. Find the area of the middle section if the areas of the outer sections are \(S_{1}\) and \(S_{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{S_1 + S_2}{2}}
Let \( K \) be a field and \( a \in K \) such that \( f(x) = x^n - a \) is irreducible. If \( m \in \mathbb{N} \) divides \( n \) and \( \alpha \) is a root of \( f(x) \) in an extension field of \( K \), find the minimal polynomial of \( \alpha^m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^{n/m} - a}
In a 12-hour interval (from 0 hours to 12 hours), how many minutes are there when the value of the hour is greater than the value of the minutes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Determine the asymptotic complexity of the recurrence relation given by $T(1) = 1$ and $T(n) = 3T(\sqrt[3]{n}) + \log_{2}(\log_{2}n)$. Express your answer using big O notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(\log n)}
Compute the area of an ideal triangle in the Poincaré disk model, where the triangle's vertices are on the boundary of the disk. One side of the triangle is a vertical geodesic that intersects a semicircle at the points (0,0) and (1,0). The semicircle is described by the equation \((x-\frac{1}{2})^2 + y^2 = \frac{1}{4}\) with \(y > 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Jude bought three chairs for his house, all at the same price. He also bought a table, four lamps, two sets of plates at $20 for each set, and a painting. The table costs twice as much as one of the chairs, and the lamps each cost half of the table's price. The painting originally costs $100 but was 25% off. After giving the cashier $350, Jude got a change. How much did each of the chairs cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$26}
Determine the number of solutions \((\alpha, \beta)\) in the finite field \(F_{q^2}\) such that \(\alpha + \alpha^q + \beta \beta^q = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q^3}
Identify the smallest non-abelian group that is not isomorphic to a cyclic group, a direct product of cyclic groups, or a semi-direct product of cyclic groups, excluding finite abelian groups and groups of order up to 7.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Q_8}
Given that $X$ follows a uniform distribution $Uniform(1, 9.9)$ and, conditionally on $X = x$, $Y$ is normally distributed $Normal(1.4, x^2)$, find the covariance between $X$ and $Y$, denoted as $Cov(X, Y)$. Calculate $E[XY]$ as part of your solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out you must open all the doors and disarm all the mines. In the room is a panel with $3$ buttons, which conveniently contains an instruction manual. The red button arms a mine, the yellow button disarms two mines and closes a door, and the green button opens two doors. Initially $3$ doors are closed and $3$ mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
From a point on a circle, two chords are drawn with lengths of 10 cm and 12 cm. Find the radius of the circle if the distance from the midpoint of the shorter chord to the longer chord is 4 cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.25 \text{ cm}}
Determine the number of elements of order \( p \) in the group \( \text{GL}(2, \mathbb{Z}_p) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2 - 1}
Given three complex numbers $(z_1, z_2, z_3)$ lying on the unit circle and related by the equation $ \frac{ z_1^2}{z_2 z_3} + \frac{ z_2^2}{z_1 z_3} + \frac{ z_3^2}{z_1 z_2} +1 =0$, find the sum of all possible values of $|z_1 + z_2 + z_3|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A rectangle ABCD is given. On the side AB, \( n \) different points are chosen strictly between A and B. Similarly, \( m \) different points are chosen on the side AD between A and D. Lines are drawn from the points parallel to the sides. How many rectangles are formed in this way?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(m+2)(m+1)(n+2)(n+1)}{4}}
考虑1,2···，20的排列$( a _ { 1 } a _ { 2 } \cdots a _ { 2 0 } )$，对此排列进行如下操作：对换其中某两个数所处的位置，目标是将此排列变为（1,2，···，20）．设对每一个排列$a = ( a _ { 1 } , a _ { 2 } , \cdots ,$ $a _ { 2 0 } )$达到目标所需要进行操作次数的最小值记为$k _ { a }$，求$k _ { a }$的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
If $\sqrt{x+3}$ is the simplest quadratic radical, then the possible value of $x$ is ________. (Fill in the correct answer)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Observe the following equations: $a + b = 1$, $a^2 + b^2 = 3$, $a^3 + b^3 = 4$, $a^4 + b^4 = 7$, $a^5 + b^5 = 11$. Identify the pattern in the sequence of results on the right side of the equations and calculate $a^{10} + b^{10}$ directly. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{123}
Calculate the partial derivative of the function \( f(A) = \langle Ax, By \rangle \) with respect to the matrix \( H \), where \( A, B \in L(\mathbb{R}^n, \mathbb{R}^k) \), \( x, y \in \mathbb{R}^n \), and \( H \in L(\mathbb{R}^n, \mathbb{R}^k) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{B y x^\top}
Let \( a_{1}, a_{2}, \cdots, a_{2014} \) be a permutation of the positive integers \( 1, 2, \cdots, 2014 \). Define
\[ S_{k} = a_{1} + a_{2} + \cdots + a_{k} \quad (k=1, 2, \cdots, 2014). \]

What is the maximum number of odd numbers among \( S_{1}, S_{2}, \cdots, S_{2014} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1511}
In triangle \( \triangle ABC \), \( BC = 4 \) and \( \angle BAC = 60^\circ \). Let \( I \) be the incentre of \( \triangle ABC \). The circle passing through \( B, I \), and \( C \) meets the perpendicular bisector of \( BC \) at a point \( X \) inside \( \triangle ABC \). Find \( AX \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4 \sqrt{3}}{3}}
Let \( \text{Revolution}(x) = x^3 + Ux^2 + Sx + A \), where \( U \), \( S \), and \( A \) are all integers and \( U + S + A + 1 = 1773 \). Given that \( \text{Revolution} \) has exactly two distinct nonzero integer roots \( G \) and \( B \), find the minimum value of \( |GB| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{392}
Darnell pays $12 for unlimited texting, calling, and data on his phone each month. An alternative phone plan would charge $1 per 30 texts, $3 per 20 minutes of calls, $5 per 2GB of data usage, and an additional $2 per month for international calls. Darnell sends 60 texts, spends 60 minutes on the phone, uses 3GB of data, and makes 10 minutes of international calls each month. How many dollars less would he pay on the alternative plan?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$11}
A semicircle with radius 2021 has diameter $AB$ and center $O$. Points $C$ and $D$ lie on the semicircle such that $\angle AOC < \angle AOD = 90^{\circ}$. A circle of radius $r$ is inscribed in the sector bounded by $OA$ and $OC$ and is tangent to the semicircle at $E$. If $CD=CE$, compute $\lfloor r \rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{673}
Determine the number of $6 \times 6$ Latin squares that do not contain a $3 \times 2$ saturated sub-matrix (SSM).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of terms required in the Taylor series expansion of $\arctan x$ around $0$ to approximate $\pi$ to 10 decimal places when $x = 1$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40000000000}
Evaluate the limit: \[ \lim_{n \to \infty} \left\{ \frac{n!}{\prod_{k=1}^{n} H_k} \right\}^{\frac{1}{n}} \] where \( H_k = \sum_{i=1}^{k} \frac{1}{i} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given the function $f\left( x \right)=\sin \left( \omega x+\varphi \right)\left( \omega > 0,\left| \varphi \right|\leqslant \dfrac{\pi }{2} \right)$, and knowing that $x=-\dfrac{\pi }{4}$ is a zero point of $f\left( x \right)$, $x=\dfrac{\pi }{4}$ is the axis of symmetry of the graph of $y=f\left( x \right)$, and $f\left( x \right)$ is monotonic in $\left( \dfrac{\pi }{18},\dfrac{5\pi }{36} \right)$, find the maximum value of $\omega$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the radius of convergence of the power series \( \sum_{n=0}^\infty n!z^{n!} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
$f(0)=0, f(1) = 1$, and for every $n \geq 1$, $f(3n-1)= f(n)-1, f(3n)= f(n), f(3n+1)=f(n)+1$. So $f(2011)$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3
}
Find $10110_2\times10100_2\div10_2$. Express your answer in base 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11011100_2}
Consider the paths from \((0,0)\) to \((6,3)\) that only take steps of unit length up and right. Compute the sum of the areas bounded by the path, the \(x\)-axis, and the line \(x=6\) over all such paths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{756}
Given that $x, y, z$ are positive real numbers satisfying $\frac{1}{3} \le xy + yz + zx \le 3$, determine the range of possible values for the product $xyz$. Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1]}
Compute the Riemann-Stieltjes integral \( \int_{-1}^1 \frac{1}{1+x^2} \, dg(x) \), where \( g(x) = \begin{cases} 1 & \text{if } x<0, \\ 0 & \text{if } x=0, \\ -1 & \text{if } x>0. \end{cases} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Given the vector \( \boldsymbol{a}_{k}=\left(\cos \frac{k \pi}{6}, \sin \frac{k \pi}{6}+\cos \frac{k \pi}{6}\right) \), where \( k=0,1,\cdots,12 \), find the value of \( \sum_{k=0}^{11} \boldsymbol{a}_{k} \cdot \boldsymbol{a}_{k+1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\sqrt{3}}