
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 pentagon is formed by joining, in order, the points $(0,0)$, $(1,2)$, $(3,2)$, $(4,0)$, and $(2,-1)$ back to $(0,0)$. The perimeter of the pentagon can be written in the form $a + b\sqrt{2} + c\sqrt{5} + d\sqrt{10}$, where $a$, $b$, $c$, and $d$ are whole numbers. Find $a+b+c+d$.

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