
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 a regular triangular pyramid \( P-ABC \), where \( AB = 1 \) and \( AP = 2 \), the plane \( \alpha \) passing through \( AB \) bisects its volume. What is the cosine of the angle between edge \( PC \) and plane \( \alpha \)?

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