
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 eccentricity $\frac{5}{3}$, and the distance from point $A(a,0)$ to the asymptotes is $\frac{12}{5}$. <br/>$(1)$ Find the equation of the hyperbola; <br/>$(2)$ If the line $l$ intersects the hyperbola $C$ at points $M$ and $N$, and $\overrightarrow{AM} \cdot \overrightarrow{AN}=0$, investigate whether the line $l$ passes through a fixed point. If it passes through a fixed point, find the coordinates of that point; if not, explain the reason.

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