
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
The sides of triangle $PQR$ are in the ratio $3:4:5$. Segment $PS$ is the angle bisector drawn to the shortest side, dividing it into segments $QS$ and $SR$. What is the length, in inches, of the longer subsegment of side $QR$ if the length of side $QR$ is $12$ inches?

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