
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 that circle $C$ passes through points $A(1,1)$ and $B(-2,-2)$, and line $m$: $2x-y=4$ bisects circle $C$.

(1) Find the equation of circle $C$;

(2) If circle $C$ intersects with line $x-y+a=0$ at points $A$ and $B$, and $OA \perp OB$, where $O$ is the origin, find the value of the real number $a$.

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