
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 polygon is formed by joining the points $(0,0)$, $(1,1)$, $(3,1)$, $(3,2)$, $(2,2)$, $(2,0)$, and back to $(0,0)$ in order. The perimeter of this polygon can be expressed as $a + b\sqrt{2} + c\sqrt{3}$, where $a$, $b$, and $c$ are integers. Calculate $a+b+c$.

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