
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
(1) In the Cartesian coordinate system $xOy$, the focal length of the hyperbola $\frac{x^{2}}{7}-\frac{y^{2}}{3}=1$ is ______.

(2) The solution set of the inequality $\frac{x-2}{x-1}\geqslant 2$ is ______.

(3) Given $F_{1}(-4,0)$, $F_{2}(4,0)$, a moving point $M$ satisfies $|MF_{1}|+|MF_{2}|=10$, then the trajectory equation of the moving point $M$ is ______.

(4) Given the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$, a point $P$ on the left branch is at a distance of $10$ from the left focus, then the distance from point $P$ to the right focus is ______.

(5) If the sequence $\{a_{n}\}$ satisfies $a_{1}=1$, $a_{1}+2a_{2}+3a_{3}+\ldots+na_{n}=n^{2}a_{n}$, then $a_{2017}=$ ______.

(6) Given $m$, $n\in\mathbb{R}^{+}$, $\frac{1}{m}+ \frac{1}{n}=2$, when $m+n$ takes the minimum value $2$, $m$, $n$ correspond to the midpoint of a chord of the hyperbola $\frac{x^{2}}{4}- \frac{y^{2}}{2}=1$, then the equation of the line containing this chord is ______.

Present the answer in LaTex format: \boxed{Your answer}
user