Problem Description

Given the pseudocode below to find a target number I between $1$ and r $($both Inclusive$)$:

while $(1)\{$

int $-$ i $*$ d $=(1+$ r$)/2$ if $($i mid$)$ r $=$ pi$*$id $-$ t else if $($i $>=$ id$)1$mid$+1$; else $1$ print$("$Found$")$;

break:

$1$

You are required to answer q queries, where in each query, you will be provided with values $1$ and r For each query, determine the following:

What is the probability that the above code will fall to print "Found" when any value i $($where l $<-$ i $<$ r $)$ is chosen? Express this probability as an integer where the fraction SP$/$Q and gcd$($P$.$ Q$)1.$ You should compute P $=$ Q $^$ lambda $-1$ modulo $1019\mathrm{.}7$ where QA$-1$ denotes the multiplicative inverse of a modulo $108947$

Problem Constraints f$=$A$\backslash |||0|$