Hyperdimensional Array Notation

Hyperdimensional Array Notation (HDAN) is a notation defined by SeveralLegend9998.SeveralLegend9998's Googology Series (Retrieved 23 October, 2022.) Unfortunately, the original definition of this notation is ill-defined by the reason explained in #Issues section.
=="Definition"==
Unfortunately, it's too hard to give the main definitions (And it'd be too complicated if done so).
Here are some tips that can be "helpful":
#\(a\underbrace{\#(1)(1)(1)...(1)(1)(1)}_{c}b = a\underbrace{\#(1)(1)(1)...(1)(1)(1)}_{c-1}ab\) where \(\#\) indicates the rest of the expression.
#\(a(2)b = a^b\)
#\(a\underbrace{(1)(1)(1)...(1)(1)(1)}_{c}(2)b = a^{\underbrace{aaaaa...aaaaa}_{c \text{ a's}}b}\)
#\(a\underbrace{(b)(b)(b)...(b)(b)(b)}_{c}d = \{\underbrace{a,a,a,a,a,...a,a,a,a,a}_{b -1 \text{ a's}},d,c\}\)
#\(a(1,b)c = a(a(1,b)d,b-1)a\) where \(a(b,c)1 = a\), \(a(\#,1)b = a(\#)b\) and d = c-1
#\(a(b,c)d = a\underbrace{(b-1,c)(b-1,c)(b-1,c)...(b-1,c)(b-1,c)(b-1,c)}_{d\text{ (b-1,c)'s}}a\)
==Issues==
The definition of the function was ill-defined as some complicated expressions (e.g. 3(5)(2)(3,2)(4)(7)4) are not formalised.
==Sources==
Category:Notations
Category:Unformalised
Category:Incomplete
Category:Array Notations