
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 $F$ is a common focus of the ellipse $C_{1}$: $\frac{x^2}{4}+\frac{y^2}{3}=1$ and the parabola $C_{2}$: $y^{2}=2px$ $(p>0)$, $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$. Calculate the length of segment $AB$.

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