# They call it Height and Distance problem?

Geometry Level 3

$$P$$ is top and $$Q$$ is the foot of a tower standing on a horizontal plane. $$A$$ is the foot a building standing beside the tower on the same horizontal plane, $$B$$ is another point on the building such that $$AB$$ is $$32 \text{ m}$$. It is found that,

$\begin{cases} \cot(\angle PAQ)=\dfrac{2}{5} \\ \cot(\angle PBQ)=\dfrac{3}{5}\end{cases}$

If the height of tower (in meters) is in the form $$a\sqrt{b}-c$$ where $$a,b,c \in \mathbb{N}$$ and $$b$$ is square-free, then find $$a+b+c$$.

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