
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\\),  
\\((1)\\) find the equation of the line on which the common chord \\(AB\\) lies;  
\\((2)\\) find the equation of the circle whose center lies on the line \\(y=-x\\) and passes through points \\(A\\) and \\(B\\);  
\\((3)\\) find the equation of the circle that passes through points \\(A\\) and \\(B\\) and has the smallest area.

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