Let \( a\uparrow\uparrow b = a^{a^{\cdots^{a}}} \), where there are \( b \) instances of \( a \). The operation \( a\uparrow\uparrow b \) is defined recursively as follows:  
\[  a\uparrow\uparrow b = \begin{cases} a & \text{if } b=1 \\ a^{a\uparrow\uparrow (b-1)} & \text{if } b\ge2 \end{cases} \]  
Determine the remainder of \( 3\uparrow\uparrow( 3\uparrow\uparrow ( 3\uparrow\uparrow 3)) \) when divided by \( 60 \).

Let's think step by step and output the final answer within \boxed{}.
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