
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
We cut out an equilateral triangle from a homogeneous sheet. We suspend it at a certain point $P$ on its perimeter, a distance $x$ from one of its vertices. The median divides the triangle into two parts; in what ratio are the masses of these two parts? When are these two masses equal? When does the ratio of the two masses assume an extreme value, and what is this extreme value?

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