
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 $\overrightarrow{AB} \perp \overrightarrow{AC}$, $|\overrightarrow{AB}|= \frac{1}{t}$, $|\overrightarrow{AC}|=t$, and point $P$ is a point on the plane of $\triangle ABC$ such that $\overrightarrow{AP}= \frac{\overrightarrow{AB}}{|\overrightarrow{AB}|} + \frac{4\overrightarrow{AC}}{|\overrightarrow{AC}|}$. Find the real value(s) of $t$ that satisfy $\overrightarrow{AP} \perp \overrightarrow{BC}$.

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