
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  $S$  be the sum of all positive integers that can be expressed in the form  $2^a \cdot 3^b \cdot 5^c$ , where  $a$ ,  $b$ ,  $c$  are positive integers that satisfy  $a+b+c=10$ . Find the remainder when  $S$  is divided by  $1001$ .

*Proposed by Michael Ren*

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