# I don't have all day!

Algebra Level 3

$1 \# ( 2\# (3\# (4\# \cdots (98 \# 99 )))\cdots )$

Let $$\#$$ denote a binary operator such that $$\color{red}{a}\# \color{blue}{b} =\color{red}{a}+\color{blue}{b}+\color{red}{a}\color{blue}{b}$$. Then what is the value of the expression above?

Notation: $$n!$$ denote the factorial of $$n$$.

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