
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
Two players,  $B$  and  $R$ , play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with  $B$ . On  $B$ 's turn,  $B$  selects one white unit square and colors it blue. On  $R$ 's turn,  $R$  selects two white unit squares and colors them red. The players alternate until  $B$  decides to end the game. At this point,  $B$  gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score  $B$  can guarantee?

(A *simple polygon* is a polygon (not necessarily convex) that does not intersect itself and has no holes.)

*Proposed by David Torres*

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