
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
 $P$  is a point inside the triangle  $ABC$ . The line through  $P$  parallel to  $AB$  meets  $AC$   $A_0$  and  $BC$  at  $B_0$ . Similarly, the line through  $P$  parallel to  $CA$  meets  $AB$  at  $A_1$  and  $BC$  at  $C_1$ , and the line through P parallel to BC meets  $AB$  at  $B_2$  and  $AC$  at  $C_2$ . Find the point  $P$  such that  $A_0B_0 = A_1B_1 = A_2C_2$ .

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