
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 passes through the points $(1,2)$, $(-3,2)$, and $(-1,2\sqrt{2})$.

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

$(2)$ If the chord $AB$ passing through point $P(-1,2)$ has a length of $2\sqrt{7}$, find the equation of line $AB$.

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