Nested Right Triangle

Round 19

Statement

    Let $\triangle ABC$ be a right-angled triangle with $\angle ACB = 90^\circ.$ The inscribed circle $\odot O$ of $\triangle ABC$ is tangent to $BC, CA , AB$ at $D, E, F$ respectively. $AD$ intersects $\odot O$ at $P,$ $\angle BPC = 90^\circ.$

    Given that $CD=6,$ length of segment $AD$ can be written as $m\sqrt {n} + m$. Find out the value of $m+n$
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