
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
Points  $P_1, P_2, P_3,$  and  $P_4$  are  $(0,0), (10, 20), (5, 15),$  and  $(12, -6)$ , respectively. For what point  $P \in \mathbb{R}^2$  is the sum of the distances from  $P$  to the other  $4$  points minimal?

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