
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 $A = (0, 10)$ and $B = (b, 12)$ be points on the coordinate plane. Let $PQRSTU$ be a convex equilateral hexagon such that $\angle UAP = 120^\circ,$ $\overline{AB}\parallel \overline{QR},$ $\overline{BC}\parallel \overline{RS},$ $\overline{CD}\parallel \overline{UV},$ and the y-coordinates of its vertices are distinct elements of the set $\{10, 12, 14, 16, 18, 20\}$. Find the area of the hexagon.

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