
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
Triangle  $ABC$  has side lengths  $AB=120$ ,  $BC=220$ , and  $AC=180$ . Lines  $\ell_{A}$ ,  $\ell_{B}$ , and  $\ell_{C}$  are drawn parallel to  $\overline{BC}$ ,  $\overline{AC}$ , and  $\overline{AB}$ , respectively, such that the intersection of  $\ell_{A}$ ,  $\ell_{B}$ , and  $\ell_{C}$  with the interior of  $\triangle ABC$  are segments of length  $55$ ,  $45$ , and  $15$ , respectively. Find the perimeter of the triangle whose sides lie on  $\ell_{A}$ ,  $\ell_{B}$ , and  $\ell_{C}$ .

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