A geometry problem by Danish Ahmed

In triangle $PQR$ , $PQ = QR$ and $RS$ is the altitude. $PR$ is extended to point $T$ such that $QT = 10$ .

The values of $\tan \angle RQT, \tan \angle SQT$ and $\tan \angle PQT$ form a geometric sequence, and:

The values of $\cot \angle SQT, \cot \angle RQT$ and $\cot \angle SQR$ form an arithmetic sequence.

If the area of the triangle $PQR$ can be exprssed as $\dfrac{a}{b}$ where $gcd(a ,b) = 1$ .

Then $a + b =$