
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
Farmer John ties his goat to a number of ropes of varying lengths in the Euclidean plane. If he ties the goat to  $k$  ropes centered at  $Q_1$ ,  $Q_2$ , ...  $Q_k$  with lengths  $\ell_1$ ,  $\ell_2$ , ...  $\ell_k$  (respectively), the goat can reach any point  $R$  such that  $\ell_j \geq RQ_j$  for all  $j \in \{1,2,3, \ldots k\}$ .

Suppose that Farmer John has planted grass at a finite set of points  $P_1$ ,  $P_2$ , ...  $P_n$ , and sets the ropes such that the goat can reach all of these points. What is, in terms of the points, the largest possible lower bound on the area of the region that the goat can reach?

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