inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $\Gamma_1$ be a circle with radius $\frac{5}{2}$. Points $A$, $B$, and $C$ are on $\Gamma_1$ such that $\overline{AB} = 3$ and $\overline{AC} = 5$. Let $\Gamma_2$ be a circle tangent to $AB$ and $BC$ at $Q$ and $R$, and also internally tangent to $\Gamma_1$ at $P$. $\Gamma_2$ intersects $AC$ at $X$ and $Y$. The area $[PXY]$ can be expressed as $\frac{a\sqrt{b}}{c}$. Find $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
The population of a town is 15000. It decreases by 20% in the first year due to job losses. In the second year, the population increases by 10% due to new job opportunities, but also experiences a 5% decrease due to migration. In the third year, the population increases by 8% as a result of increased birth rates, but also decreases by 25% due to an economic recession. What will be the net population of the town after these three years?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10157}
Given the Markov chain $\{X_n\}$ with transition probability matrix $P$:

$$P=\left( \begin{array}{ccccccc}
\frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 & 0 \\
0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 \\
\frac{1}{4} & 0 & 0 & \frac{1}{2} & \frac{1}{4} & 0 & 0 \\
0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\
0 & 0 & 0 & 0 & \frac{1}{2} & \frac{1}{4} & \frac{1}{4} \\
0 & 0 & 0 & 0 & \frac{1}{4} & \frac{1}{2} & \frac{1}{4} \\
0 & 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 0 \\
\end{array} \right),$$

where $X_0 = 1$, calculate the expected value of the sum $\sum_{k=1}^{T} \mathbb{I}_{\{X_k=3\}}$, where $T = \inf\{k\geq 0: X_k \in \{5,6,7\}\}$. Here, $\mathbb{I}_A$ denotes the indicator function that is $1$ when the event $A$ occurs and $0$ otherwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Given the permutations \( \theta = (13)(14)(567) \), \( \tau = (137)(2456) \), and \( \sigma = (13456)(27) \) in the symmetric group \( S_7 \), find the intersection of the left cosets \( \sigma \langle \theta \rangle \) and \( \tau \langle \theta \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Determine the total number of 7-bit strings such that any pair of these strings differ in exactly 4 bits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Jeremy is driving 600 miles to visit his parents. His average speed for this trip is 50 miles per hour. Every two hours of driving, he makes a rest stop for 15 minutes. His car gets 18 miles per gallon of gas, and his gas indicator comes on when he's used 15 gallons, at which point he stops to refill, which takes 10 minutes. How long, in minutes, does he take to get to his parents' house, if he starts out with a full tank of gas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{815 \, \text{minutes}}
A ring x of diameter 16 inches is placed inside another ring y of diameter 18 inches. Additionally, a third ring z with a diameter of 20 inches is placed outside of ring y. Calculate the combined uncovered fractions of the surface area of ring x and the surface area of ring z not covered by the rings y and z, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.19}
An unpainted cone has radius \( 3 \mathrm{~cm} \) and slant height \( 5 \mathrm{~cm} \). The cone is placed in a container of paint. With the cone's circular base resting flat on the bottom of the container, the depth of the paint in the container is \( 2 \mathrm{~cm} \). When the cone is removed, its circular base and the lower portion of its lateral surface are covered in paint. The fraction of the total surface area of the cone that is covered in paint can be written as \( \frac{p}{q} \) where \( p \) and \( q \) are positive integers with no common divisor larger than 1. What is the value of \( p+q \)?

(The lateral surface of a cone is its external surface not including the circular base. A cone with radius \( r \), height \( h \), and slant height \( s \) has lateral surface area equal to \( \pi r s \).)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59}
Given a positive integer \( n\left( {n > {2004}}\right) \) , we put 1, 2, 3, …, \( {n}^{2} \) into squares of an \( n \times  n \) chessboard with one number in a square. A
 square is called a "good square" if the square satisfies following conditions:
 There are at least 2004 squares that are in the same row with the square such that any number within these 2004 squares is
 less than the number within the square.
 There are at least 2004 squares that are in the same column with the square such that any number within these 2004 squares
 is less than the number within the square.
 Find the maximum value of the number of the "good square".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$n^2 - 2004n$
}
Find the slope of the curve of intersection of the surface \(z = x^{3} + xy^{2}\) and the plane \(2x-2y = 1\) at the point \((1, \frac{1}{2}, \frac{5}{4})\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{17}{4}}
Find a complex function that outputs only real values, similar to the function \( f(z) = |z| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(z) = |z|^2}
Determine the order of the poles of the function \( f(z)= \dfrac{z}{ (1-e^{z}) \sin z} \) at \( z=2\pi ik \), where \( k \in \mathbb{Z} \) and \( k \neq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The integer part and decimal part of $(\sqrt{10}+3)^{2n+1}$ for $n \in \mathbf{N}$ are denoted as $I$ and $F$, respectively. What is the value of $P(I + F)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In triangle $ABC$, where all sides are different, the bisector of angle $BAC$ intersects side $BC$ at point $D$.

It is known that $\angle AB - BD = a$ and $AC + CD = b$.

Find $AD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{ab}}
A dog is being treated for fleas. The effectiveness of each flea treatment varies as follows: the first treatment eliminates 60% of fleas, the second treatment eliminates 45% of the remaining fleas, the third treatment eliminates 30% of the remaining fleas, and the fourth treatment eliminates 20% of the remaining fleas. After four treatments, the dog has 20 fleas left. How many more fleas did the dog have before the flea treatments than after?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{142}
Every month Ivan pays a fixed amount of his salary for a mortgage, and the remaining portion of his salary is spent on current expenses. In December, Ivan paid 40% of his salary for the mortgage. In January, Ivan's salary increased by 9%. By what percentage did the amount spent on current expenses increase in January compared to December?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15\%}
Given point \( A \) has coordinates \( (0,3) \), and points \( B \) and \( C \) are on the circle \( O: x^{2}+y^{2}=25 \), satisfying \(\angle B A C=90^{\circ}\). Find the maximum area of \(\triangle A B C\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{25 + 3 \sqrt{41}}{2}}
对于一个实数 x, 令 $\{x\}=x-[x]$. 记 $S=\int_{0}^{2024} \min \left(\left\{\frac{x}{8}\right\},\left\{\frac{x}{4}\right\}\right) \mathrm{d} x$, 则 $[S]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{759}
The first doughnut is priced at $1 and then if you purchase additional doughnuts as dozens, the price is a certain amount per dozen. The total number of doughnuts purchased for $24 is 48. What is the price per dozen for additional doughnuts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$5.87}
Simplify the expression: \[ \frac{(8+2\sqrt{15})^{3/2}+(8-2\sqrt{15})^{3/2}}{(12+2\sqrt{35})^{3/2}-(12-2\sqrt{35})^{3/2}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{13}}
Compute the Hausdorff measure \( \mathcal{H}_{2} \) of the unit cube \( A = \{(x,y,z) \in \mathbb{R}^{3}: x\in [0,1], y \in[0,1], z\in[0,1]\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the largest constant $C$ so that for all real numbers $x$, $y$, and $z$,
\[x^2 + y^2 + z^3 + 1 \ge C(x + y + z).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Find the distance between the foci of the ellipse
\[\frac{x^2}{48} + \frac{y^2}{12} = 3.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{3}}
On March 12, 2016, the fourth Beijing Agriculture Carnival opened in Changping. The event was divided into seven sections: "Three Pavilions, Two Gardens, One Belt, and One Valley." The "Three Pavilions" refer to the Boutique Agriculture Pavilion, the Creative Agriculture Pavilion, and the Smart Agriculture Pavilion; the "Two Gardens" refer to the Theme Carnival Park and the Agricultural Experience Park; the "One Belt" refers to the Strawberry Leisure Experience Belt; and the "One Valley" refers to the Yanshou Ecological Sightseeing Valley. Due to limited time, a group of students plans to visit "One Pavilion, One Garden, One Belt, and One Valley." How many different routes can they take for their visit? (Answer with a number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
In an arithmetic sequence \(a_1, a_2, \ldots, a_{1990}\) with a non-zero common difference, where \(a_i a_j > 0\) for all \(i, j = 1, 2, \ldots, 1990\), let \(b_1 = a_1 \cdot a_{1990}\), \(b_2 = a_2 \cdot a_{1989}\), \(\ldots\), \(b_k = a_k \cdot a_{1990 - k + 1}\) for \(k = 1, 2, \ldots, 1990\). Determine the largest term among \(\{b_k\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b_{995}}
Given four independent random variables $X_1, X_2, X_3, X_4$ each uniformly distributed on the interval $(0,1)$, let $(X_{(1)}, X_{(2)}, X_{(3)}, X_{(4)})$ denote the order statistics. Compute the probability $P(X_{(2)} \leq 3X_{(1)})$. Use integration techniques to find your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Given the differential equations \( f'(x) = -\frac{f(x)}{\sqrt{(f(x))^2+(g(x))^2}} \) and \( g'(x) = 1 - \frac{g(x)}{\sqrt{(f(x))^2+(g(x))^2}} \) for all \( x \in \mathbb{R} \), with initial conditions \( g(0) = 0 \) and \( f(0) = 10 \), and knowing that \( \lim_{x\to\infty}f(x) = 0 \), find the value of \( \lim_{x\to\infty}g(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A child builds towers using identically shaped cubes of different colors. Determine the number of different towers with a height of 6 cubes that can be built with 3 yellow cubes, 3 purple cubes, and 2 orange cubes (Two cubes will be left out).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{350}
Determine the set of limit points of the set \( A = \left\{ \frac{\sqrt{m} - \sqrt{n}}{\sqrt{m} + \sqrt{n}} \mid m, n \in \mathbb{N} \right\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1]}
Evaluate the limit: $$\lim\limits_{x \to 0^{\large +}}\mrm{Li}_{x}\pars{1}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Let $A=\{1,2,\ldots,2012\}, \: B=\{1,2,\ldots,19\}$ and $S$ be the set of all subsets of $A.$ Find the number of functions $f : S\to B$ satisfying $f(A_1\cap A_2)=\min\{f(A_1),f(A_2)\}$ for all $A_1, A_2 \in S.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1^{2012} + 2^{2012} + \ldots + 19^{2012}}
Given a semicircle of diameter $AB$ and center $O$. Let $CD$ be the chord of the semicircle tangent to two circles of diameters $AO$ and $OB$. If $CD=120$ cm,, caclulate area of the semicircle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$2400 \pi \; \text{cm}^2$}
Calculate \(142^{381} \mod 773\) using the given modular arithmetic relations: \(142^1 \equiv 142 \pmod{773}\), \(142^2 \equiv 66 \pmod{773}\), and \(142^3 \equiv 96 \pmod{773}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{178}
Provide an example of a continuous function $f(x)$ such that the integral $\int_{0}^{\infty}f(x)\,dx$ converges, but the limit $\lim_{x\to\infty}f(x)$ does not exist.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin(x^2)}
Evaluate the integral \( \int_0^5 (x^2+1) \, d \lfloor x \rfloor \), where \( \lfloor \cdot \rfloor \) is the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Lei Lei bought some goats and sheep. If she had bought 2 more goats, the average price of each sheep would increase by 60 yuan. If she had bought 2 fewer goats, the average price of each sheep would decrease by 90 yuan. Lei Lei bought $\qquad$ sheep in total.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Add the square of the smallest area from squares of size $1 \times 1, 2 \times 2,$ and $3 \times 3,$ such that the number of squares of each size is the same.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let $R$ be the set of points $(x, y)$ such that $x$ and $y$ are positive, $x + y$ is at most 2013, and
\[
  \lceil x \rceil \lfloor y \rfloor = \lfloor x \rfloor \lceil y \rceil.
\]
Compute the area of set $R$.  Recall that $\lfloor a \rfloor$ is the greatest integer that is less than or equal to $a$, and $\lceil a \rceil$ is the least integer that is greater than or equal to $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006.5}
Let $I$ be an uncountable directed set, and let $T = \{E_{\alpha}: \alpha \in I\}$ be a collection of countably infinite sets such that $E_{\alpha} \subset E_{\beta}$ whenever $\alpha \le \beta$. The collection $T$ is totally ordered with respect to set inclusion. Determine the cardinality of $\bigcup_{\alpha \in I} E_{\alpha}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_1}
In triangle $ABC$, $B=\frac{π}{3}$, the angle bisector of $\angle BAC$ intersects $BC$ at $D$, and $AD=3$, $BD=2$, then $\cos C=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{6}-1}{6}}
Find the smallest natural number $k$ such that for all sufficiently large $n$, less than 1% of all permutations of $[n]$ have at least $k$ fixed points. Assume that the number of derangements $D(n)$ for a set of $n$ elements satisfies $D(n) = n! \sum_{k=0}^n \frac{(-1)^k}{k!}$ and $\sum_{k=0}^{\infty} \frac{(-1)^k}{k!} = \frac{1}{e}$. Provide your answer as a single integer $k$. \(\boxed{}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $A B C D$ be an isosceles trapezoid with parallel bases $A B=1$ and $C D=2$ and height 1. Find the area of the region containing all points inside $A B C D$ whose projections onto the four sides of the trapezoid lie on the segments formed by $A B, B C, C D$ and $D A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{8}}
A store purchased 3 different types of electronics: Radios, TVs, and Mobile Phones in the following quantities and cost prices:

- 5 Radios at a cost price of Rs. 8,000 each.
- 3 TVs at a cost price of Rs. 20,000 each.
- 4 Mobile Phones at a cost price of Rs. 15,000 each.

The store offers different discounts and sales tax rates on each type of electronic:

- Radios: 10% discount and 6% sales tax on the price after discount.
- TVs: 15% discount and 7% sales tax on the price after discount.
- Mobile Phones: 8% discount and 5% sales tax on the price after discount.

The store sells all the electronic items and receives the following sale amounts post-discount and including sales tax:

- Radios: Rs. 7,200 each.
- TVs: Rs. 18,000 each.
- Mobile Phones: Rs. 14,500 each.

Calculate the combined loss percentage of the store on selling all these electronic items.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.5\%}
In Vila Par, all the truth coins weigh an even number of grams, while the false coins weigh an odd number of grams. The electronic device only provides the parity (odd or even) of the total weight of a set of coins. If there are $2020$ truth coins and $2$ false coins, determine the smallest integer $k$ such that there exists a strategy to identify the two false coins using the electronic device at most $k$ times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Find the area of the portion of the plane \( z = x + 3y \) that lies inside the elliptical cylinder given by the equation \( \frac{x^2}{4} + \frac{y^2}{9} = 1. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\pi\sqrt{11}}
Determine the number of subgroups of index $p$ in the additive group $\mathbb{Z}^2$, where $p$ is a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p + 1}
Given a positive integer $k$, let \|k\| denote the absolute difference between $k$ and the nearest perfect square. For example, \|13\|=3 since the nearest perfect square to 13 is 16. Compute the smallest positive integer $n$ such that $\frac{\|1\|+\|2\|+\cdots+\|n\|}{n}=100$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89800}
From a solid regular tetrahedron with edge length of one unit, a smaller regular tetrahedron with an edge length of $\frac{1}{n}$ is cut off from each of its vertices, where $n$ is an integer greater than 1. For what value of $n$ will the ratio of the surface area to the volume of the remaining solid be minimized?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Which prime numbers can be expressed as the sum of two positive composite numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
A caravan of camels, 1 km long, moves uniformly through the desert. A rider travels from the end of the caravan to the front and back to the end. During this time, the caravan covers 1 km. How far did the rider travel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \sqrt{2}}
Let  $ABCD$  be a rectangle with  $AB = 6$  and  $BC = 6 \sqrt 3$ .  We construct four semicircles  $\omega_1$ ,  $\omega_2$ ,  $\omega_3$ ,  $\omega_4$  whose diameters are the segments  $AB$ ,  $BC$ ,  $CD$ ,  $DA$ . It is given that  $\omega_i$  and  $\omega_{i+1}$  intersect at some point  $X_i$  in the interior of  $ABCD$  for every  $i=1,2,3,4$  (indices taken modulo  $4$ ). Compute the square of the area of  $X_1X_2X_3X_4$ .

*Proposed by Evan Chen*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
14 Labeled as $1,2, \cdots, 100$, there are some matches in the matchboxes. If each question allows asking about the parity of the sum of matches in any 15 boxes, then to determine the parity of the number of matches in box 1, at least how many questions are needed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
1. In how many ways can a $3 \times 3$ square table be filled with the numbers $2,2,3,3,3,4,4,4,4$ so that the sum of the numbers in each $2 \times 2$ square of this table is equal to 14?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find all positive integers $k$ such that for any positive numbers $a, b, c$ that satisfy the inequality $k(a b+b c+c a)>5\left(a^{2}+b^{2}+c^{2}\right)$, there always exists a triangle with side lengths $a, b, c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Consider two fictional states: Alpha and Beta. Alpha issues license plates with a format of two letters followed by four numbers, and then ending with one letter (LLNNNNL). Beta issues plates with three letters followed by three numbers and lastly one letter (LLLNNNL). Assume all 10 digits and 26 letters are equally likely to appear in the respective slots. How many more license plates can state Alpha issue than state Beta?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{281216000}
The area of the region bounded by the graph of $x^2+y^2 = 3|x-y| + 3|x+y|$ is $m+n\pi$, where $m$ and $n$ are integers. What is the value of $m + n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
B4. On each of the 10,000 fields of a $100 \times 100$ chessboard, there is a number. On the top row, from left to right, the numbers are 0 to 99. In the left column, from top to bottom, the numbers are 0 to 99. The sum of four numbers in a $2 \times 2$ block is always 20. What number is in the bottom right corner of the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-178}
Distribute 5 students into dormitories A, B, and C, with each dormitory having at least 1 and at most 2 students. Among these, the number of different ways to distribute them without student A going to dormitory A is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
10 people -- liars and knights -- sat around a round table. Liars always lie, and knights always tell the truth. Each of them was given a coin. Then each person passed their coin to one of their two neighbors. After that, 5 people said, "I have one coin," and the remaining 5 people said, "I have no coins." What is the maximum number of knights that could have been sitting at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given that A, B, C, D are distinct integers from 1 to 6, and the curves y = Ax^2 + B and y = Cx^2 + D intersect, calculate the number of distinct ways to choose values for A, B, C, D such that the curves intersect, considering that the intersection is independent of the order of A, B, C, D. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
In the plane Cartesian coordinate system $xOy$, $AB$ is a chord of the parabola $y^{2}=4x$ passing through the point $F(1,0)$, and the circumcircle of triangle $\triangle AOB$ intersects the parabola at point $P$ (different from points $O, A, B$). If $PF$ bisects the angle $\angle APB$, find all possible values of $|PF|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{13} - 1}
In quadrilateral \(ABCD\), it is known that \(AB = BC\) and \(\angle ABC = \angle ADC = 90^{\circ}\). From vertex \(B\), a perpendicular \(BH\) is dropped to side \(AD\). Find the area of quadrilateral \(ABCD\) given that \(BH = h\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{h^2}
If the lengths of the major axis, minor axis, and focus of an ellipse form a geometric sequence, then the eccentricity of this ellipse is ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{5} - 1}{2}}
Given proposition $p$: "$1\leqslant x\leqslant 5$ is a sufficient but not necessary condition for $x^{2}-(a+1)x+a\leqslant 0$", and proposition $q$: "There are two triangles $\triangle ABC$ that satisfy $AC=6$, $BC=a$, and $\angle CAB=30^{\circ}$". If the proposition $\lnot p\land q$ is true, find the range of values for the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 < a\leqslant 5}