
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
Joshua likes to play with numbers and patterns.  Joshua's favorite number is  $6$  because it is the units digit of his birth year,  $1996$ .  Part of the reason Joshua likes the number  $6$  so much is that the powers of  $6$  all have the same units digit as they grow from  $6^1$ :
\begin{align*}6^1&=6,6^2&=36,6^3&=216,6^4&=1296,6^5&=7776,6^6&=46656,\vdots\end{align*}
However, not all units digits remain constant when exponentiated in this way.  One day Joshua asks Michael if there are simple patterns for the units digits when each one-digit integer is exponentiated in the manner above.  Michael responds, "You tell me!"  Joshua gives a disappointed look, but then Michael suggests that Joshua play around with some numbers and see what he can discover.  "See if you can find the units digit of  $2008^{2008}$ ," Michael challenges.  After a little while, Joshua finds an answer which Michael confirms is correct.  What is Joshua's correct answer (the units digit of  $2008^{2008}$ )?

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