
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 $\odot M:x^{2}+y^{2}-2x-2y-2=0$, line $l:2x+y+2=0$, $P$ is a moving point on $l$. Through point $P$, tangents $PA$ and $PB$ are drawn to circle $\odot M$, with tangents at points $A$ and $B$. Determine the equation of line $AB$ when $|PM|\cdot |AB|$ is minimized.

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