
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
A circle with center \( O \) lies in a horizontal plane. The diameter \( AB \) has length 4. Point \( P \) lies vertically above \( O \) such that \( PO = 2\sqrt{2} \). Point \( C \) lies on the semicircular arc \( AB \) such that the ratio of the lengths of the arcs \( AC \) and \( CB \) is \( 2:1 \).

What is the shortest distance from \( A \) to \( PC \)?

A) \( \sqrt{2} \)

B) \( \sqrt{3} \)

C) 2

D) \( 2\sqrt{2} \)

E) 3

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