
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 how many integer values of  $m$ ,
(i)  $1\le m \le 5000$ (ii)  $[\sqrt{m}] =[\sqrt{m+125}]$ Note:  $[x]$  is the greatest integer function

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