
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
If $a$, $b$, and $c$ are the three sides of $\triangle ABC$, and $a$, $b$ satisfy the relationship $|a-2|+(b-5)^{2}=0$, $c$ is the largest integer solution of the inequality system $\left\{{\begin{array}{l}{x-3>3(x-4)}\\{\frac{4x-1}{6}<x+1}\end{array}}\right.$, find the perimeter of $\triangle ABC$.

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