
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 $PQR$, point $S$ is on line segment $PQ$ such that $S$ divides $PQ$ in the ratio 1:3. Points $T$ and $U$ lie on $PR$ and $QR$, respectively, with $PT:TR = 1:2$ and $QU:UR = 3:1$. Let $V$ be the point where angle bisectors of $\angle PSR$ and $\angle QSU$ intersect within the triangle. Given $PQ = 30$, $PR = 20$, and $QR = 40$, find the length of $SV$.

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