
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let   ${ a\uparrow\uparrow b = {{{{{a^{a}}^a}^{\dots}}}^{a}}^{a}} $ , where there are  $ b $  a's in total. That is  $ a\uparrow\uparrow b $  is given by the recurrence \[  a\uparrow\uparrow b = \begin{cases} a & b=1 a^{a\uparrow\uparrow (b-1)} & b\ge2\end{cases} \] What is the remainder of  $  3\uparrow\uparrow( 3\uparrow\uparrow ( 3\uparrow\uparrow 3)) $  when divided by  $ 60 $ ?

Present the answer in LaTex format: \boxed{Your answer}
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