
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 a hyperbola with its foci on the $x$-axis and its real axis length is $4\sqrt{2}$, and its eccentricity $e=\frac{\sqrt{5}}{2}$.

$(1)$ Find the standard equation of the hyperbola;

$(2)$ The real axis of the hyperbola is $A_1A_2$, point $P$ is a moving point on the hyperbola, and moving point $Q$ satisfies $A_1Q \perp A_1P$, $A_2Q \perp A_2P$, find the equation of the trajectory of point $Q$.

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