
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. Given that the line $l$ passing through the point $M(-3,0)$ is intercepted by the circle $x^{2}+(y+2)^{2}=25$ to form a chord of length $8$, what is the equation of line $l$?
2. Circles $C_{1}$: $x^{2}+y^{2}+2x+8y-8=0$ and $C_{2}$: $x^{2}+y^{2}-4x-4y-2=0$ intersect. What is the length of their common chord?
3. What is the equation of a circle passing through points $A(0,2)$ and $B(-2,2)$, with its center lying on the line $x-y-2=0$?
4. Given the circle with the equation $(x-2)^{2}+y^{2}=1$, a line passing through the external point $P(3,4)$ intersects the circle at points $A$ and $B$. What is the value of $\overrightarrow{PA}\cdot \overrightarrow{PB}$?

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