
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 two points $P\_1(4,9)$ and $P\_2(6,3)$, find the equation of the circle with $P\_1P\_2$ as its diameter;
Find the equation of the circle passing through two points $A(2,-3)$ and $B(-2,-5)$, and with its center on the line $l$: $x-2y-3=0$.

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