
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  $B = (20, 14)$  and  $C = (18, 0)$  be two points in the plane. For every line  $\ell$  passing through  $B$ , we color red the foot of the perpendicular from  $C$  to  $\ell$ .  The set of red points enclose a bounded region of area  $\mathcal{A}$ . Find  $\lfloor \mathcal{A} \rfloor$  (that is, find the greatest integer not exceeding  $\mathcal A$ ).

*Proposed by Yang Liu*

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