
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 $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $\Gamma$: $\frac{{x}^{2}}{{a}^{2}}+y^{2}=1\left(a \gt 1\right)$, $A$ is the top point of $\Gamma$, and the line $l$ passes through point $F_{1}$ and intersects $\Gamma$ at points $B$ and $C$. If $l$ vertically bisects the line segment $AF_{2}$, then the perimeter of $\triangle ABC$ is ______.

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