
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 $\frac {x^{2}}{4} + \frac {y^{2}}{3} = 1$ with its left and right foci labeled as $F_1$ and $F_2$ respectively, a line $l$ passing through $F_1$ intersects the ellipse at points $A$ and $B$. Find the maximum value of $|BF_2| + |AF_2|$.

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