
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 $PQR$ has vertices at $P(8,10)$, $Q(4,0)$, and $R(9,4)$. The point $S$ with coordinates $(x,y)$ is chosen inside the triangle so that the three small triangles $PQS$, $PRS$, and $QRS$ all have equal areas. Calculate $10x + y$ and find the perimeter of triangle $PQR$.

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