Let $a_{i+1}=(c*a_{i} + d) \mod m$ where $c=5447196$, $d=1953840$ and $m=998244353$ be a pseudo random number generator for $0 \leq i \leq N-3$ where $N=100000$ and $a_0=a_{N-1}=a_{N}=0$

One day, Taniya created a game in which there were $N$ blocks arranged sequentially. A player starts at the $1st$ block. To go from the $ith$ block to the $(i+1)th$ block, the player has to take a jump. The jump will succeed with probability $p$, meaning that the player will go to the $(i+1)th$ block with probability $p$,otherwise he will have to start from the beginning and will be landed at the $1st$ block. The game ends when the player reaches the $Nth$ block. Further, there is a penalty of $a_i$ coins each time a player lands on block $i$. Taniya was not sure what value to pick for $p$ and so she randomly selected a number from $[0.5\ ,\ 1]$. Calculate the Expected number of coins a player has to pay before the game ends.

It can be shown that the answer can be expressed as $\frac{p}{q}$, where $\gcd(p,q) = 1$ and $q \neq 0 \mod m$. Output $pq^{-1} \mod m$.