
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 a positive real number \( k \), for any positive real numbers \( a \) and \( b \), let \( m = \min \left\{ a, \frac{b}{k a^2 + b^2} \right\} \). Then the maximum value of \( m \) is \(\quad \). (Note: \(\min \{ x, y \} \) represents the smaller of the real numbers \( x \) and \( y \)).

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