
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
Find the maximal possible value of  $n$  such that there exist points  $P_1,P_2,P_3,\ldots,P_n$  in the plane and real numbers  $r_1,r_2,\ldots,r_n$  such that the distance between any two different points  $P_i$  and  $P_j$  is  $r_i+r_j$ .

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