
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 the circle $M: x^2 + y^2 + 2mx - 3 = 0$ (where $m < 0$) has a radius of $2$, and the left focus of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{3} = 1$ is $F(-c, 0)$, find the value of $a$.

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