
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 $P = (0,0)$ and $Q = (q,3)$ be points on the coordinate plane. Let $PQRSTUV$ be a convex equilateral hexagon such that $\angle VPQ = 120^\circ,$ $\overline{PQ}\parallel \overline{ST},$ $\overline{QR}\parallel \overline{TU,}$ $\overline{RS}\parallel \overline{UV},$ and the y-coordinates of its vertices are distinct elements of the set $\{0,3,6,9,12,15\}.$ The area of the hexagon can be written in the form $x\sqrt {y},$ where $x$ and $y$ are positive integers and y is not divisible by the square of any prime. Find $x + y.$

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