
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
The original population of Nosuch Junction was a perfect square. If it was increased by $100$, the population was one more than a perfect square. However, after an additional increase of $100$, the population was again a perfect square. Determine the factor that the original population is a multiple of.

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