
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
In the Cartesian coordinate system $xOy$, a circle $C$ passes through points $P(3+2\sqrt{2},0)$, $Q(3-2\sqrt{2},0)$, and $R(0,1)$.
$(1)$ Find the equation of circle $C$;
$(2)$ If circle $C$ intersects with the line $x-y+a=0$ at points $A$ and $B$, and $OA \perp OB$, find the value of $a$.

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