
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
Consider the set  $S$  of  $100$  numbers:  $1; \frac{1}{2}; \frac{1}{3}; ... ; \frac{1}{100}$ .
Any two numbers,  $a$  and  $b$ , are eliminated in  $S$ , and the number  $a+b+ab$  is added. Now, there are  $99$  numbers on  $S$ . 
After doing this operation  $99$  times, there's only  $1$  number on  $S$ . What values can this number take?

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