
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 triangles $ABC$ and $ABD$ are inscribed in the same semicircle with diameter $AB$, which measures $15 \mathrm{~cm}$. From $D$, a perpendicular is drawn to $AB$ which intersects $AB$ at $P$, the segment $AC$ at $Q$, and the extension of side $BC$ at $R$. Furthermore, $PR=\frac{40}{3} \mathrm{~cm}$ and $PQ=\frac{15}{4} \mathrm{~cm}$.

a) Verify that $PQ \cdot PR = BP \cdot AP$.

b) Find the length of the segment $DP$.

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