
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 set containing all positive integers whose decimal representations contain only 3’s and 7’s, have at most 1998 digits, and have at least one digit appear exactly 999 times. If  $N$  denotes the number of elements in  $S$ , find the remainder when  $N$  is divided by 1000.

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