
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 the right focus of the hyperbola $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, and a circle $C$ with center $F$: $x^{2}+y^{2}-4x+3=0$ is tangent to the asymptotes of the hyperbola, find the standard equation of the hyperbola.

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