
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 that $A$ and $B$ are the left and right vertices of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, $F_{1}$ and $F_{2}$ are the left and right foci, and $C$ is a point on the hyperbola different from $A$ and $B$. If the lines $AC$ and $BC$ intersect the right directrix of the hyperbola at points $M$ and $N$, respectively, find $\overrightarrow{F_{1} M} \cdot \overrightarrow{F_{2} N}$.

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