
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  $ABC$  be an equilateral triangle and let  $P_0$  be a point outside this triangle, such that  $\triangle{AP_0C}$  is an isoscele triangle with a right angle at  $P_0$ . A grasshopper starts from  $P_0$  and turns around the triangle as follows. From  $P_0$  the grasshopper jumps to  $P_1$ , which is the symmetric point of  $P_0$  with respect to  $A$ . From  $P_1$ , the grasshopper jumps to  $P_2$ , which is the symmetric point of  $P_1$  with respect to  $B$ . Then the grasshopper jumps to  $P_3$  which is the symmetric point of  $P_2$  with respect to  $C$ , and so on. Compare the distance  $P_0P_1$  and  $P_0P_n$ .  $n \in N$ .

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