
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 that point $P$ is any point on the circle $(x+1)^{2}+y^{2}=12$ with center $F_{1}$, point $F_{2}(1,0)$, and the perpendicular bisector of segment $PF_{2}$ intersects radius $PF_{1}$ at point $M$.
(1) Find the trajectory equation of moving point $M$;
(2) A line $l$ passing through point $F_{2}$ ($l$ does not coincide with the $x$-axis) intersects $M$'s trajectory at two distinct points $A$ and $B$, find the maximum value of the inradius $r$ of triangle $F_{1}AB$, and find the equation of line $l$ at this time.

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