
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 circles $C: x^{2}+y^{2}-10x-10y=0$ and $M: x^{2}+y^{2}+6x+2y-40=0$ intersect at points $A$ and $B$.

(1) Find the equation of the common chord that lies on the line connecting the intersection points of circles $C$ and $M$.

(2) Find the length of $AB$.

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