
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 moving circle is internally tangent to the circle $F\_1: (x+1)^2 + y^2 = 9$ and externally tangent to the circle $F\_2: (x-1)^2 + y^2 = 1$.
(1) Find the equation of the trajectory $L$ of the moving circle's center $M$;
(2) Let the line $l: x = my + 1$ intersect with the trajectory $L$ at points $A$ and $B$. Does the area of triangle $ABF\_1$ have a maximum value? If so, find this maximum value and the equation of the line $l$. If not, explain why.

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