
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 smallest positive real number  $r$  with the following property:  For every choice of  $2023$  unit vectors  $v_1,v_2, \dots ,v_{2023} \in \mathbb{R}^2$ , a point  $p$  can be found in the plane such that for each subset  $S$  of  $\{1,2, \dots , 2023\}$ , the sum  $$ \sum_{i \in S} v_i $$  lies inside the disc  $\{x \in \mathbb{R}^2 : ||x-p|| \leq r\}$ .

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