
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) The length of the chord intercepted by the line $y=2x+3$ on the circle $x^{2}+y^{2}-6x-8y=0$ is equal to \_\_\_\_\_\_\_\_\_.

(2) The midpoint of a chord of the circle ${{(x-1)}^{2}}+{{(y+2)}^{2}}=3$ is $(\frac{1}{2},-\frac{3}{2})$, the equation of the line containing this chord is \_\_\_\_\_\_\_.

(3) Given the circle system $x^{2}+y^{2}-2ax+2(a-2)y+2=0 (a\neq 1, a\in R)$, the circle system always passes through the fixed point \_\_\_\_\_\_\_\_.

(4) A light ray is shot out from point $A\left( -2,3 \right)$, and after being reflected by the $x$-axis, it is tangent to the circle $C:{{x}^{2}}+{{y}^{2}}-6x-4y+12=0$. The equation of the line containing the reflected light ray is \_\_\_\_\_\_\_\_\_.

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