
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  $\phi(n)$  denote  $\textit{Euler's phi function}$ , the number of integers  $1\leq i\leq n$  that are relatively prime to  $n$ .  (For example,  $\phi(6)=2$  and  $\phi(10)=4$ .)  Let \[S=\sum_{d|2008}\phi(d),\] in which  $d$  ranges through all positive divisors of  $2008$ , including  $1$  and  $2008$ .  Find the remainder when  $S$  is divided by  $1000$ .

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