
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
A line $l$ passes through the intersection point $P$ of line $l_1: 2x-y=1$ and line $l_2: x+2y=3$, and is perpendicular to line $l_3: x-y+1=0$.
(1) Find the equation of line $l$;
(2) If line $l$ is tangent to the circle $C: (x-a)^2+y^2=8$ ($a>0$), find $a$.

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