
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
Given two circles $C_{1}$: $x^{2}+y^{2}+2x+2y-8=0$ and $C_{2}$: $x^{2}+y^{2}-2x+10y-24=0$ intersect at points $A$ and $B$,<br/>$(1)$ Find the equation of the line containing the common chord $AB$;<br/>$(2)$ Find the equation of the circle with its center on the line $y=-x$ passing through points $A$ and $B$;<br/>$(3)$ Find the equation of the circle passing through points $A$ and $B$ with the smallest area.

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