
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 positive integer is called simple if its ordinary decimal representation consists entirely of zeroes and ones. Find the least positive integer  $k$  such that each positive integer  $n$  can be written as  $n = a_1 \pm a_2 \pm a_3 \pm \cdots \pm a_k$  where  $a_1, \dots , a_k$  are simple.

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