
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
An ellipse $C: \frac{x^2}{a^2} + y^2 = 1$ (with $a > 0$) has an eccentricity of $\frac{\sqrt{3}}{2}$. If $F_1$ and $F_2$ are the foci of ellipse $C$, and a line $l$ passing through $F_1$ intersects $C$ at points $A$ and $B$, then the maximum value of $|AF_2| + |BF_2|$ is ____.

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