
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 points \( F_1 \) and \( F_2 \) are the left and right foci of a hyperbola \( C \) with its center at the origin and its axes along the coordinate axes. A line \( l \) passing through point \( F_2 \) intersects the right branch of the hyperbola at points \( A \) and \( B \). Points \( I_1 \) and \( I_2 \) are the incenters of triangles \( \triangle A F_1 F_2 \) and \( \triangle B F_1 F_2 \), respectively. If the eccentricity of the hyperbola \( C \) is 2, \(\left| I_1 I_2 \right| = \frac{9}{2}\), and the sine of the angle of inclination of line \( l \) is \( \frac{8}{9} \), find the equation of the hyperbola \( C \).

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