
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 the quadrilateral pyramid $P-ABCD$, $PA \perp$ base $ABCD$, $BC=CD=2$, $AC=4$, $\angle ACB = \angle ACD = \frac{\pi}{3}$. $F$ is the midpoint of $PC$, and $AF \perp PB$.
(1) Find the length of $PA$;
(2) Find the sine of the dihedral angle $B-AF-D$.

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