
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 $C$ that passes through points $A(-1, 0)$ and $B(3, 0)$, and its center lies on the line $x - y = 0$.

1. Find the equation of circle $C$;
2. If point $P(x, y)$ is any point on circle $C$, find the maximum and minimum values of the distance from point $P$ to the line $x + 2y + 4 = 0$.

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