
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 ellipse $\dfrac{x^2}{4} + y^2 = 1$, a line $l$ passing through point $M(-1,0)$ intersects the ellipse at points $A$ and $B$, with $O$ being the origin.

$(1)$ Find the equation of the trajectory of the midpoint $P$ of segment $AB$;

$(2)$ Find the maximum area of $\triangle OAB$ and write the equation of line $l$ at this time.

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