
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
In the plane a point  $O$  is and a sequence of points  $P_1, P_2, P_3, \ldots$  are given. The distances  $OP_1, OP_2, OP_3, \ldots$  are  $r_1, r_2, r_3, \ldots$  Let  $\alpha$  satisfies  $0 < \alpha < 1.$  Suppose that for every  $n$  the distance from the point  $P_n$  to any other point of the sequence is  $\geq r^{\alpha}_n.$  Determine the exponent  $\beta$ , as large as possible such that for some  $C$  independent of  $n$  
\[r_n \geq Cn^{\beta}, n = 1,2, \ldots\]

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