
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 an ellipse \( C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) where \( a > b > 0 \), with left and right foci \( F_{1} \) and \( F_{2} \) respectively. A line \( l \) passing through \( F_{1} \) with inclination angle \( \alpha \) intersects the ellipse \( C \) at points \( A \) and \( B \). Given that \( \cos \alpha = \frac{5}{13} \), \( 5 \overrightarrow{F_{1} A} = 8 \overrightarrow{B F_{1}} \), and the distance from \( F_{2} \) to the line segment \( AB \) is \( \frac{72}{13} \): 

(1) Find the equation of the ellipse \( C \).

(2) Suppose \( P \) and \( Q \) are points on the ellipse \( C \)'s right directrix such that \( |PQ| = 3 \). Find the equation of the locus of the incenter \( M \) of triangle \( \triangle F_{1} P Q \).

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