
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  $S = \{1, 22, 333, \dots , 999999999\}$ . For how many pairs of integers  $(a, b)$  where  $a, b \in S$  and  $a < b$  is it the case that  $a$  divides  $b$ ?

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