Juca dreams of retiring and spending the rest of his life peacefully. He calculated that he needs to have at least $$$R$$$ thousand reais. Juca has a variable income job, and at the end of each year, he receives a payment that varies from $$$1$$$ to $$$R$$$ thousand reais, with uniform probability.

If, after receiving his salary, his total income is $$$R$$$ thousand reais or more, he retires happily. Otherwise, he decides to adjust his income according to inflation:

• Let $$$S$$$ be the balance of his account.
• Let $$$a, b$$$ be two integers that satisfy $$$0 \leq a \lt b$$$, representing the level of inflation.
• The new balance will be (floor(S * (1 - a / b))), where floor is the floor function, which returns the integer part of a positive real number. For example, if $$$S = 3, a=1,b=2$$$, the new balance would be floor($$$1.5$$$) = $$$1$$$.

Juca is curious to know the expected value of the number of years it will take for him to retire. It can be shown that the answer is a fraction $$$\frac{p}{q}$$$ with $$$gcd(p,q) = 1$$$. Print $$$p \cdot q^{-1} \mod (10^9 + 7)$$$.