
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
Let \( P \) be the vertex of a cone, and \( A, B, C \) be three points on the circumference of its base circle, such that \(\angle ABC = 90^\circ\). Let \( M \) be the midpoint of \( AP \). Given that \( AB = 1 \), \( AC = 2 \), and \( AP = \sqrt{2} \), determine the dihedral angle \( M-BC-A \).

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