
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 hyperbola $C$: $\frac{b^{2}x^{2}}{a^{2}}-\frac{a^{2}y^{2}}{b^{2}}=a^{2}b^{2} (a > 0, b > 0)$, a circle $x^{2}+y^{2}=a^{2}$ is tangent to the left focus $F_{1}$ of the hyperbola at point $T$, and intersects the right branch of the hyperbola at point $P$. If point $T$ is the midpoint of segment $F_{1}P$, find the equations of the asymptotes of the hyperbola.

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