
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
Cut a regular dodecagon into squares and equilateral triangles.

Let \( P_{1}, P_{2}, \ldots, P_{12} \) be the consecutive vertices of a regular dodecagon. What can be said about the intersection of the diagonals \( P_{1} P_{9}, P_{2} P_{11} \), and \( P_{4} P_{12} \)?

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