
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 $C$ passes through points $A(-2,0)$, $B(0,2)$, and the center of the circle is on the line $y=x$. Also, the line $l:y=kx+1$ intersects circle $C$ at points $P$ and $Q$.

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

$(2)$ If $\overrightarrow{OP} \cdot \overrightarrow{OQ} = -2$, find the value of the real number $k$.

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