
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 $E: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ with its right vertex at $A(2,0)$, a line $l$ passes through the point $P(4,0)$. When the line $l$ intersects the hyperbola $E$ at exactly one point, the distance from point $A$ to line $l$ is $\frac{2\sqrt{5}}{5}$.  
$(1)$ Find the standard equation of the hyperbola $E$;  
$(2)$ If the line $l$ intersects the hyperbola $E$ at points $M$ and $N$, and there exists a point $Q(t,0)$ on the $x$-axis such that $\angle MQP = \angle NQP$ always holds, find the value of $t$.

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