
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 a circle $C$ whose center is on the right branch of the hyperbola $E$: $x^{2}- \frac {y^{2}}{3}=1$, and the circle $C$ passes through the right focus $F$ of the hyperbola $E$, and is tangent to the line $x=-2$, find the length of the segment cut by circle $C$ on the $x$-axis.

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