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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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