
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 a circle with center $C$ passing through points $A(1,1)$ and $B(2,-2)$, and the center $C$ lies on the line $l: x+y+5=0$.
(1) Find the equation of circle $C$;
(2) If the chord passing through point $D(-1,-1)$ on line $m$ cuts circle $C$ into a chord length of $2\sqrt {21}$, find the equation of line $m$.

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