
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 circle C: $x^2+y^2-8x+12=0$, and line l: $x+ay+2a=0$.
(1) Find the value of $a$ when line $l$ is tangent to circle $C$;
(2) When line $l$ intersects circle $C$ at points $A$ and $B$, and $|AB|=2\sqrt{2}$, find the equation of line $l$.

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