
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
If  $f(n)$  denotes the number of divisors of  $2024^{2024}$  that are either less than  $n$  or share at least one prime factor with  $n$ , find the remainder when   $$ \sum^{2024^{2024}}_{n=1} f(n) $$   is divided by  $1000$ .

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