
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
Let $F$ be the right focus of the hyperbola $E$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 (a>0, b>0)$, $A$ and $B$ be the left and right vertices of the hyperbola $E$, respectively. Point $P$ is a moving point on the hyperbola $E$ other than $A$ and $B$. The line $l: x=t$ intersects the line perpendicular to $AP$ passing through $F$ at point $Q$. Given that points $B$, $P$, and $Q$ are collinear, find the maximum value of $\frac{t}{a}$.

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