
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
Given a point $P$ inside the triangle $\triangle ABC$, let $P_1, P_2, P_3$ be the reflections of $P$ over the sides $AB, BC,$ and $CA$ respectively. Then:

1. \( P_1 P_2 = PB \sqrt{2(1-\cos 2B)} \), \( P_2 P_3 = PC \sqrt{2(1-\cos 2C)} \), \( P_3 P_1 = PA \sqrt{2(1-\cos 2A)} \).
2. \( \angle P_1 P_2 P_3 = \angle BPC - \angle A \), \( \angle P_2 P_3 P_1 = \angle CPA - \angle B \), \( \angle P_3 P_1 P_2 = \angle APB - \angle C \).

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