
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
Let  $\theta$  be an obtuse angle with  $\sin{\theta}=\frac{3}{5}$ . If an ant starts at the origin and repeatedly moves  $1$  unit and turns by an angle of  $\theta$ , there exists a region  $R$  in the plane such that for every point  $P\in R$  and every constant  $c>0$ , the ant is within a distance  $c$  of  $P$  at some point in time (so the ant gets arbitrarily close to every point in the set). What is the largest possible area of  $R$ ?

*2020 CCA Math Bonanza Individual Round #15*

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