Suppose that we have symbols \(a,b,c,d,e\) and an operation \(\clubsuit\) on the symbols satisfying the following rules:
\(x\;\clubsuit\;e = x\), where \(x\) can be any of \(a,b,c,d,e\).
\(a\;\clubsuit\;c = c\;\clubsuit\;a = b\;\clubsuit\;d = d\;\clubsuit\;b = e\).
any bracketing of the same string of symbols are the same; for example, \(((a\;\clubsuit\;c)\;\clubsuit\;d)\;\clubsuit\;(a\;\clubsuit\;d) = (a\;\clubsuit\;(c\;\clubsuit\;(d\;\clubsuit\;(a\;\clubsuit\;d))))\).
\((a\;\clubsuit\;b)\clubsuit\;c = d\).
We use the power notation. If \(n\geq 1\) is a natural number, we write \(a^n\) for \((\dots(a\;\clubsuit\;a)\;\clubsuit\dots)\;\clubsuit\; a\), where \(a\) appears \(n\) times. Similarly for other symbols. Let \(p,q,r,s\geq 1\) be natural numbers. Express \(a^p\;\clubsuit\;b^q\;\clubsuit\;a^r\;\clubsuit\;b^s\) using the symbols \(a,b,c,d\) no more than once (power notation allowed).