
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
The line \( l \) is the left directrix of the ellipse \(\frac{x^2}{4} + \frac{y^2}{3} = 1\), \( F_1 \) is the left focus, point \( P \) moves along \( l \). The point \( M \) is the third vertex of an equilateral triangle with side \( PF_1 \), with points \( P \), \( F_1 \), and \( M \) arranged counterclockwise. Find the equation of the locus of point \( M \).

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