
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
(1) 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.

(2) Given line $l$: $x - y + 3 = 0$ is intercepted by circle $C$: $(x - a)^2 + (y - 2)^2 = 4 (a > 0)$ to form a chord with length $2\sqrt{2}$, find the value of $a$.

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