
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
Fill integers from $1$ to $10000$ in natural order into a $100 \times 100$ table, with each row filling from left to right: the first row is $1$ to $100$, the second row is $101$ to $200$, and so forth, with the hundredth row being $9901$ to $10000$. If the row and column containing the number $A$ do not contain any perfect squares, then $A$ is called a free number. How many free numbers are there in the table?

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