
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
For a binary string  $S$  (i.e. a string of 0 's and 1's) that contains at least one 0 , we produce a binary string  $f(S)$  as follows:
- If the substring 110 occurs in  $S$ , replace each instance of 110 with 01 to produce  $f(S)$ ;
- Otherwise, replace the leftmost occurrence of 0 in  $S$  by 1 to produce  $f(S)$ .
Given binary string  $S$  of length  $n$ , we define the lifetime of  $S$  to be the number of times  $f$  can be applied to  $S$  until the resulting string contains no more 0 's. For example, $$ 

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