
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
A hexagon is formed by joining, in order, points $(0,0)$, $(1,1)$, $(2,1)$, $(3,0)$, $(2,-1)$, $(1,-1)$, and back to $(0,0)$. Calculate the perimeter of this hexagon and represent it in the form $$a+b\sqrt{2} + c\sqrt{3}$$ where $a$, $b$, and $c$ are integers. Find the sum $a + b + c$.

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