
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
Consider triangle $PQR$ where angle $P$'s bisector intersects side $QR$ at point $S$. A line segment starts at vertex $P$ with zero length and moves linearly towards vertex $R$. As it moves, its length increases until it is perpendicular to $PS$ and then starts to decrease until it reaches $R$ with length zero again. Determine the maximum length of the segment during its movement.
A) $\frac{2ab}{a+b}$
B) $\frac{ab}{\sqrt{a^2 + b^2}}$
C) $\frac{2ab}{\sqrt{a^2 + b^2}}$
D) $\frac{a+b}{2}$

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