
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
Find the standard equation of the circle that satisfies the following conditions:
- $(1)$ The center of the circle is $(1,-3)$ and it passes through the point $(-3,-1)$.
- $(2)$ The center of the circle lies on the line $x=2$ and intersects the $y$-axis at points $A(0,-4)$ and $B(0,-2)$.

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