inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $X_t$ be a stochastic process. Using the rules of Ito's calculus, find the value of $\langle dX_t, dX_t \rangle$ when $dX_t = Y_t \, dt$, where $Y_t$ is a progressive stochastic process.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the sequence defined by the initial conditions $a_0=a_1=a_2=a_3=a_4=0$ and $a_5=1$, and the recurrence relation $$a_{n+6} = \frac{a_{n+5} + a_{n+4} + a_{n+3} + a_{n+2} + a_{n+1} + a_{n}}{6},$$ find the limit of $a_n$ as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{7}}
Find the value of \( f(f(100)) \) given the function \( f(x) = ([\{x\}] + 3 - x^7)^{1/7} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Find the Ramsey number for the graphs $(m+1)K_4$ and $K_4$, given that $R(mK_4,K_4) + 4 \leq R((m+1)K_4,K_4)$ for $m \geq 2$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4m + 18}
Determine the number of subfields of the splitting field of the polynomial \(x^{42} - 1\) over \(\mathbb{Q}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Misha rolls a standard, fair six-sided die until she rolls $1$-$2$-$3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\tfrac{m}{n}$, 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{647}
Consider a Markov chain with a transition probability matrix $P$ where each transition probability is 0.5, except for the transitions from the first and last states, which have a probability of 1. Calculate the expected number of steps required to reach state $j$ starting from state 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(j-1)^2}
Consider a 100-dimensional hypersphere with a radius of 4000 miles. What fraction of its volume lies within 40 miles of its surface?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.634}
Find the number of ordered pairs of integers $(a,b)\in\{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a "quadratic residue modulo $x^2+1$ and $35$", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following $\textit{equivalent}$ conditions holds:
there exist polynomials $P$, $Q$ with integer coefficients such that $f(x)^2-(ax+b)=(x^2+1)P(x)+35Q(x)$;
or more conceptually, the remainder when (the polynomial) $f(x)^2-(ax+b)$ is divided by (the polynomial) $x^2+1$ is a polynomial with integer coefficients all divisible by $35$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
225
}
Determine the smallest non-abelian group $G$ such that the power map $f(x) = x^n$ is a homomorphism for some integer $n > 2$. What is this group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_3}
Calculate the supremum of \( \int_{-1}^1 f \, dg \) for \( f \in \mathcal{F} \), where \( g(x) = x^2 \) on \([-1, 1]\) and \( \mathcal{F} = \{ f \in \mathcal{C}[-1,1] : |f(x)| \leq 5 \text{ for all } x \in [-1,1] \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Calculate the flux of the vector field \( F(x,y,z) = (x,y,z) \) across the surface defined by \( x^2 - y^2 + z^2 = 0 \) with \( y \in [0, 1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the volume enclosed between the surfaces \(x+y+2z=2\) and \(2x+y+z=4\) in the first octant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{10}{3}}
Given a polynomial \( P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \ldots + a_1 x + a_0 \) where each coefficient \( a_i \) belongs to the interval \([100,101]\), determine the smallest natural number \( n \) such that this polynomial can have a real root.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Find the value of \(\alpha\) such that the roots of the polynomial \(x^5 + 5\lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8\lambda + 3)x + \lambda\alpha - 2 = 0\) are independent of \(\lambda\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Find the value of \( b \) such that the tangent line to the function \( f_a(x) = (x-a)e^{a+2-x} \) at \( x = b \) intersects the y-axis at the point \( A(0, 2012) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
For given integers $x$ and $d$, determine the largest number of $\mathbb{Z}^d$ lattice points that are mutually $x$-visible. Denote this number as $g(x,d)$. For instance, it is known that $g(0,2) = 4$ and $g(1,2) \ge 9$. What is $g(x,d)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x + 2)^d}
Find the interval of convergence for the power series with coefficients \(a_n= \frac{(\ln n)^2}{n}.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1)}
King Arthur has two advisors, Merlin and Percival, each providing correct answers with probability \( p \). If both advisors agree, the king follows their advice; if they disagree, he flips a coin. Arthur considers firing one advisor if it doesn't decrease the probability of making correct decisions. Merlin, aiming to avoid being fired, will give an answer that maximizes his chance of staying. What is the probability that Merlin will be fired?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given that the function $f(x,y)$ is differentiable at $(1,1)$ and $f(1,1)=\frac{\partial f}{\partial x}(1,1)=\frac{\partial f}{\partial y}(1,1)$, find the value of $\frac{d}{dx}f(x,f(x,x))$ when $x=1$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine if there is a particular solution to the equation $x = \log(x) + c$ without using numerical methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c \geq 1}
Find the real number \( x \) represented by the continued fraction \([12;2,2,12,2,2,12,2,2,12\dots]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 + \sqrt{41}}
Given a matroid $M$ with a ground set $E$ of size $2n$, where there exists a subset $A \subseteq E$ of size $n$ such that both $A$ and $E \setminus A$ are independent, determine the minimum number of subsets $B \subseteq E$ such that both $B$ and $E \setminus B$ are independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n}
Let $A$ be the matrix defined as follows:

$$A := \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

Determine the number of matrices similar to $A$ over the field $\mathbb{Z}/3\mathbb{Z}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{117}
Find an element of order 3 in the multiplicative group \( \mathbb{Z}_{67}^* \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Given positive real numbers $x$, $y$, and $z$ such that $\frac{1}{3} \leq xy + yz + zx \leq 3$, find the range of possible values for $x + y + z$. Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, \infty)}
Question 150: Four spheres with a radius of 1 are pairwise externally tangent. Then the volume of the circumscribed regular tetrahedron is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\sqrt{3}(6+\\sqrt{6})^{3}}{27}"]}
Find the center of gravity of a hemispherical shell with radius $a$ and uniform density of 1, positioned with its base on the $x-y$ plane. Use the method of slicing parallel to the $x-y$ plane to determine the center of gravity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{a}{2}}
The monotonically decreasing interval of the function $f(x)= \sqrt {3-2x-x^{2}}$ is $\boxed{\text{answer}}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1,1]}
Triangle $PQR$ has vertices $P(0, 10)$, $Q(3, 0)$, $R(9, 0)$. A line through $Q$ bisects the area of $\triangle PQR$. Find the sum of the slope and $y$-intercept of this line.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{-20}{3}}
Determine the values of \( n \) for which it is possible to assemble a cube with an edge length of \( 2n \) cm from smaller cubes obtained by dividing a solid cube with an edge length of \( n \) cm into 1 cm edge length cubes. The assembled cube may have a hollow interior.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 23}
在正四面体 P-ABC 中，点 E,F 分别在棱 PB, PC 上. 若 PE\neq PF，且 AE=AF=2, EF=\sqrt{3}，则四面体 P-AEF 的体积为__________.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{9}
Given six students – three boys and three girls – standing in a row, where girls A, B, and C are not allowed to stand next to each other and girls are not allowed to stand at the ends of the row, calculate the number of different arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Equilateral triangles \(ABC\) and \(A_1B_1C_1\) with side length 10 are inscribed in the same circle such that point \(A_1\) lies on arc \(BC\) and point \(B_1\) lies on arc \(AC\). Find \(AA_1^2 + BC_1^2 + CB_1^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
There are 8 distinct points on a plane, where no three are collinear. An ant starts at one of the points, then walks in a straight line to each one of the other points, visiting each point exactly once and stopping at the final point. This creates a trail of 7 line segments. What is the maximum number of times the ant can cross its own path as it walks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A circle with radius $r$ is contained within the region bounded by a circle with radius $R$. The area bounded by the larger circle is $\frac{a}{b}$ times the area of the region outside the smaller circle and inside the larger circle. Then $R:r$ equals:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt{a}:\sqrt{a-b}$}
If the coordinates of point $N$ are $\left(a,2a-1\right)$, then point $N$ must not be in the ________ quadrant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{second}
What is the name of the operation $U^TRU$, where $U$ is an isometric embedding from $\mathbb{R}^d$ to $\mathbb{R}^n$ with $d<n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{compression}
There are 40 students in Jimin's class. Jimin and Jungkook's math scores are 98 and 100, respectively. Excluding these two students, the average score in math is 79. What is the average math score of Jimin's class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
In the orchard, 170 apples, 268 tangerines, and 120 mangoes were loaded equally into each of trucks. There are 8 apples left, 2 tangerines short, and 12 mangoes left. Find the maximum number of trucks in the orchard.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
How many multiplications and subtractions are required in the Cholesky decomposition of an $n \times n$ matrix, excluding square roots and divisions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{n^3-n}{6}\)}
Let $M$ and $N$ be two closed subspaces of a Hilbert space $H$ such that $N \subset M$. If $P_M$ and $P_N$ are the orthogonal projections onto $M$ and $N$, respectively, determine the subspace of $H$ on which $P_M - P_N$ is an orthogonal projection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M \ominus N}
Dave had 15 apps and some files on his phone. After deleting some apps and files he had 21 apps and 4 files left. Dave has 17 more apps than files left on his phone. How many files did Dave initially have on his phone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Calculate the flux of the vector field \( \bar{V}=z\bar{i}+x\bar{j}+y\bar{k} \) through the portion of the sphere \( x^2+y^2+z^2=1 \) that lies in the first octant using Gauss's theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
To combat the mice, Leopold the Cat needs to drink a potion called "ozverin" daily. He has a bottle of ozverin solution in water (a colorless, clear liquid) in the closet with a volume of \( V = 0.5 \) liters and an initial ozverin concentration of \( C_{0} = 40\% \). Each day, Leopold drinks \( v = 50 \) ml of the solution, and to avoid detection by the mice, he refills the bottle with the same volume of water. Find the volume concentration of ozverin in the bottle after the cat has drunk \( n = 5 \) times \( v = 50 \) ml of the solution and refilled the bottle with water each time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23.6 \%}
There are $25$ towns in a country. Find the smallest $k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:
1) From each town, there are exactly $k$ direct routes to $k$ other towns.
2) If two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A jar contains quarters (worth $0.25$ each), nickels (worth $0.10$ each), and pennies (worth $0.02$ each). The value of the quarters is $12.50$, the value of the nickels is $15.00$, and the value of the pennies is $8.00$. If Lena randomly chooses one coin from the jar, what is the probability that it is a quarter?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Given a regular tetrahedron \(P-ABC\) with edge length 1, let \(E\), \(F\), \(G\), and \(H\) be points on edges \(PA\), \(PB\), \(CA\), and \(CB\) respectively. If \(\overrightarrow{EF}+\overrightarrow{GH}=\overrightarrow{AB}\) and \(\overrightarrow{EH} \cdot \overrightarrow{FG}=\frac{1}{18}\), then find \(\overrightarrow{EG} \cdot \overrightarrow{FH}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{18}}
When a cube is cut into two pieces with a single plane cut, two polyhedra are obtained. Determine the number of faces of a polyhedron that cannot be obtained in this way. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A cubic polynomial $p(x)$ satisfies
\[p(n) = \frac{1}{n^2}\]for $n = 1, 2, 3,$ and $4.$  Find $p(5).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{5}{12}}
What could be the minimum length of a cycle in a graph where no single vertex is connected to all others, any two non-adjacent vertices have a common neighbor, and if the number of vertices is denoted by $n$, the sum of the squares of the degrees is $n^2 - n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the integral \( \frac{1}{2 \pi} \int_0^{2 \pi} \frac{1 - r^2}{1 - 2r \cos(\theta) +r^2}d\theta \) for \(0 < r < 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the representing object of the functor $D_n : \mathsf{Cat} \to \mathsf{Set}$, which maps a category to the set of all composable $n$-tuples of morphisms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[n]}
Determine the values of $r$ for which the Joukowsky transform is injective on the circle of radius $r$ centered at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r \neq 1}
A frog is at the point \((0,0)\). Every second, he can jump one unit either up or right. He can only move to points \((x, y)\) where \(x\) and \(y\) are not both odd. How many ways can he get to the point \((8,14)\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{330}
Bob is building raised beds for his vegetable garden. Each bed has a certain height, 2 feet wide, and 8 feet long. The sides are going to be built of 1-foot wide planks. Bob needs 50 8-foot long planks to construct 10 raised beds. What is the height of each raised bed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.25}
Compute the directional derivative of the function \( f(x,y) = \begin{cases} \frac{x^3}{x^2+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} \) at the point \((0,0)\) in the direction of the unit vector \( u = (u_1, u_2)^T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{u_1^3}
Determine the Galois group of the polynomial \( f(x) = x^2 + 2ix + 1 \) over \( \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{e\}}
Find the value of the integral $\int_0^\infty \frac{\ln ^2z} {1+z^2}{d}z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi^3}{8}$}
Evaluate the integral \( \displaystyle\int_{C(1,2)}\frac{\cos z}{z^3}\,\mathrm{d}z \), where \( C(1,2) \) is the circle centered at 1 with radius 2, using the Cauchy integral formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi i}
A hotel consists of a $2 \times 8$ square grid of rooms, each occupied by one guest. All the guests are uncomfortable, so each guest would like to move to one of the adjoining rooms (horizontally or vertically). Of course, they should do this simultaneously, in such a way that each room will again have one guest. In how many different ways can they collectively move?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1156}
Evaluate \( \lim_{x \to 0^+} \sum_{n=1}^\infty \sin(\sqrt{n}) e^{-nx} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
On a circle with a circumference of 2013 units, there are 2013 equally spaced points, each with a token. The distance between two points is the length of the shorter arc between them. Determine the largest integer $n$ such that the tokens can be rearranged so that each point still has a token, and the distance between any two tokens, originally separated by no more than $n$ points, increases.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{670}
Evaluate the triple integral: $$\int_{0}^{2}\int_{6}^{1}\int_{0}^{1} xyz\sqrt{2-y^2-x^2}\,dx\,dy\,dz$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8\sqrt{2} - 4}{15}}
A bee starts flying from point $P_0$. She flies 1 inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$, she is exactly $a\sqrt{b} + c\sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$, and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. Find the value of $a+b+c+d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2024}
In \(\triangle ABC\), if \(\tan A\), \(\tan B\), \(\tan C\) are in harmonic progression, find the minimum value of \(\cot \frac{B}{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Vasya plays the strings of a 6-string guitar from 1 to 6 and then back. Each subsequent hit lands on an adjacent string. On which string number will the 2000th hit land? (The order of string hits is: $1-2-3-4-5-6-5-4-3-2-1-2-\ldots)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Pål has more chickens than he can manage to keep track of. Therefore, he keeps an index card for each chicken. He keeps the cards in ten boxes, each of which has room for $2021$ cards.

Unfortunately, Pål is quite disorganized, so he may lose some of his boxes. Therefore, he makes several copies of each card and distributes them among different boxes, so that even if he can only find seven boxes, no matter which seven, these seven boxes taken together will contain at least one card for each of his chickens.

What is the largest number of chickens Pål can keep track of using this system?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5052}
Calculate the value of \( \int_{-\infty}^{\infty} y(t) \, dt \), where \( y(t) = \int_{-\infty}^{t} g(\tau) \, d\tau \) and for a function \( g(t) \), it is given that \( \int_{-\infty}^{\infty} g(t)e^{-j\omega t} \, dt = \omega e^{-2\omega^2} \) for any real value \( \omega \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-j}
How many people are needed in a room to have a 50% chance that three people share the same birthday, assuming birthdays are equally distributed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88}
An equilateral triangle and a square are inscribed in a circle as shown. $ABC$ is isosceles. The triangle and square share a common vertex. What is the number of degrees in the measure of the angle indicated by the question mark?

[asy]

import markers; defaultpen(linewidth(0.8));

pair A,B,C,D,E,F,G;

draw(unitcircle);

A=(-1,0);
B=(0,-1);
C=(1,0);
D=(0,1);

draw(A--B--C--D--A);

E=(-0.9,-0.45);
F=(0.9,-0.45);

draw(D--E--F--D);

G=(-0.76,-0.23);

markangle(Label("?"),n=1,radius=10,D,G,A,marker(stickframe(n=0),true));

label("$A$",(-0.5,-0.34));
label("$B$",(0.5,-0.34));
label("$C$",B,S);

[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Given that \(7^{-1} \equiv 55 \pmod{102}\), find \(49^{-1} \pmod{102}\), as a residue modulo 102. (Give an answer between 0 and 101, inclusive.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Given two non-negative integers $m>n$, let's say that $m$ ends in $n$ if we can get $n$ by erasing some digits (from left to right) in the decimal representation of $m$. For example, 329 ends in 29, and also in 9.
Determine how many three-digit numbers end in the product of their digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
103
}
Determine the number of digits in the duodecimal (base 12) representation of \(\tau\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find an upper bound, independent of \( p \), for the sum \( S_p = \sum_{j=1}^p \frac{p+1}{p-j+1} \frac{1}{2^j} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{3}}
In the board game Tsuro, each tile has 8 entry points, with each point connecting to exactly one other point. Given that there are 35 unique tiles, determine the number of distinct configurations of these tiles, considering rotations as equivalent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}