
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 the line $l:y=x$ intersects the circle $C:(x-a)^{2}+(y-1)^{2}=r^{2}$ (where $a$ is an integer and $r$ is a positive integer) at points $A$ and $B$, and $|AB|=2\sqrt{2}$, find the possible values of $a$.

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