
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
A repunit is a positive integer, all of whose digits are 1's. Let \( a_{1} < a_{2} < a_{3} < \ldots \) be a list of all the positive integers that can be expressed as the sum of distinct repunits. Compute \( a_{111} \).

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