
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 $A = (0,0)$ and $B = (b,1)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 150^\circ$, $\overline{AB}\parallel \overline{DE}$, $\overline{BC}\parallel \overline{EF}$, $\overline{CD}\parallel \overline{FA}$, and the y-coordinates of its vertices are distinct elements of the set $\{0,1,3,5,7,9\}$. The area of the hexagon can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m + n$.

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