# The Impossibly Long... Blackboard

Legend has it that my school has a really long blackboard. A reaaaally long blackboard. (I know, that's a really mundane way to start off a question. Please don't judge me.)

On this board, I write out the decimal representation of all the integers from 1 to $$10^{100}$$, and it takes $$0.01$$ seconds to write each digit; the total time in seconds is $$T$$.

I then choose two numbers at random $$a$$ and $$b$$ and replace both with a single number $$a+ab+b$$. After some number of iterations, the single number left is of the form $$(n+1)!-1$$.

Find

$\lfloor T \rfloor \pmod{\log_{10}(n)}$

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