
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 hyperbola \( C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (\(a>0, b>0\)) with left and right foci \( F_{1}, F_{2} \) and eccentricity \(\frac{5}{3}\), a line \( l \) passing through \( F_{1} \) is tangent to the circle \( x^{2}+y^{2}=a^{2} \) at point \( T \). The line \( l \) also intersects the right branch of the hyperbola \( C \) at point \( P \). Find \(\frac{\left|\overrightarrow{F_{1} P}\right|}{\left|\overrightarrow{F_{1} T}\right|} \).

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