
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
Let \( F_{1} \) and \( F_{2} \) be the left and right foci of the hyperbola \( C: \frac{x^{2}}{4}-\frac{y^{2}}{5}=1 \), respectively. Point \( P \) is on the right branch of the hyperbola \( C \). In \(\triangle P F_{1} F_{2}\), the excenter corresponding to the angle \(\angle P F_{1} F_{2}\) is \( I \), and the line \( P I \) intersects the \( x \)-axis at point \( Q \). Find the value of
\[ 
\frac{|P Q|}{|P I|}+\frac{\left|F_{1} Q\right|}{\left|F_{1} P\right|}. 
\]

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