
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
 $100$  distinct natural numbers  $a_1, a_2, a_3, \ldots, a_{100}$  are written on the board. Then, under each number  $a_i$ , someone wrote a number  $b_i$ , such that  $b_i$  is the sum of  $a_i$  and the greatest common factor of the other  $99$  numbers. What is the least possible number of distinct natural numbers that can be among  $b_1, b_2, b_3, \ldots, b_{100}$ ?

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