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Algebra Level 3

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

Let #\# denote a binary operator such that a#b=a+b+ab \color{#D61F06}{a}\# \color{#3D99F6}{b} =\color{#D61F06}{a}+\color{#3D99F6}{b}+\color{#D61F06}{a}\color{#3D99F6}{b}. Then what is the value of the expression above?

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

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