
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
Consider  $6$  points located at  $P_0=(0,0), P_1=(0,4), P_2=(4,0), P_3=(-2,-2), P_4=(3,3), P_5=(5,5)$ . Let  $R$  be the region consisting of *all* points in the plane whose distance from  $P_0$  is smaller than that from any other  $P_i$ ,  $i=1,2,3,4,5$ . Find the perimeter of the region  $R$ .

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