
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 line $l: x-y+3=0$ and a circle $C: (x-a)^2+(y-2)^2=4$ with $a > 0$, the chord intercepted on the circle by the line $l$ has a length of $2\sqrt{2}$.  
(1) Find the value of $a$.  
(2) Find the equation of the tangent line to circle $C$ that passes through point $(3,5)$.

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