
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
In triangle $\triangle ABC$, $AC=2$, $\angle BAC=\frac{π}{3}$, and $P$ is a point inside $\triangle ABC$ such that $AP\bot CP$ and $\angle APB=\frac{2π}{3}$. <br/>$(1)$ If $AP=PC$, find the area of $\triangle ABC$; <br/>$(2)$ If $BC=\sqrt{7}$, find $AP$.

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