
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

On the sides $PQ, QR, RP$ of triangle $PQR$, segments $AB, CD, EF$ are laid out. Inside the triangle, there is a given point $S_{0}$. Find the locus of points $S$ lying inside the triangle $PQR$ for which the sum of the areas of the triangles $SAB, SCD, SEF$ is equal to the sum of the areas of the triangles $S_{0}AB, S_{0}CD, S_{0}EF$. Consider the special case when
$$
\frac{AB}{PQ}=\frac{CD}{QR}=\frac{EF}{RP}.
$$

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