
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
Given the inequality:
\[ \left(\frac{1}{m+n}\right)^{2} \leqslant k\left(\frac{1}{m^{2}}+\frac{1}{n^{2}}\right) \]
for all positive integers \( m \) and \( n \), find the minimum value of the real number \( k \).

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