
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
How many pairs of natural numbers ($m, n$), each not exceeding 1000, exist such that

\[ \frac{m}{n+1} < \sqrt{2} < \frac{m+1}{n} ? \]

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