
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 parabola $C: y^{2}=2px\left(p \gt 0\right)$ with the focus $F$, a line $l$ inclined at an angle of $\frac{π}{6}$ intersects $C$ at points $A$ and $B$, and intersects the directrix of $C$ at point $M$. If $|OM|=\frac{2\sqrt{21}}{3}$ (where $O$ is the origin), calculate the length of segment $AB$.

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