
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  $(a_n)$  be a sequence of integers, with  $a_1 = 1$  and for evert integer  $n \ge 1$ ,  $a_{2n} = a_n + 1$  and  $a_{2n+1} = 10a_n$ . How many times  $111$  appears on this sequence?

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