
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
In $\triangle ABC$, $AB= 500$, $BC=550$, and $AC=650$. An interior point $P$ is drawn, and segments are drawn through $P$ parallel to the sides of the triangle such that these segments split each side into segments that are proportional to the overall sides of the triangle. Determine the length of each segment parallel to the sides, labeled as $d$.

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