
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
Suppose \( m > 0 \). If for any set of positive numbers \( a, b, c \) that satisfy \( a b c \leq \frac{1}{4} \) and \( \frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} < m \), there exists a triangle with side lengths \( a, b, c \). Determine the maximum value of the real number \( m \) and explain the reasoning.

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