Consider the sequence $\{1,3,13,31,...\}$ that is obtained by following diagonally the following array of numbers in a spiral. Find the number in the $100$th position of that sequence.

The incircle of triangle $\vartriangle ABC$ is tangent to side $AB$ at point $P$ and to side $BC$ at point $Q$. The circle passing through points $A,P,Q$ intersects line $BC$ a second time at $M$ and the circle passes through the points $C,P,Q$ and cuts the line $AB$ a second time at point$ N$. Prove that $NM$ is tangent to the incircle of $ABC$.