
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 the line $l: x=my+2 (m \in \mathbb{R})$ intersects the $x$-axis at a point which is one of the foci of the ellipse $C: \frac{x^2}{a^2} + y^2 = 1 (a > 0)$.

$(1)$ Find the equation of the ellipse $C$.

$(2)$ If the line $l$ intersects the ellipse $C$ at points $A$ and $B$, and the left focus of the ellipse $C$ is $F_1$, is there an $m$ that maximizes the area of $\triangle ABF_1$? If it exists, find the value of $m$; if not, explain why.

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