
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 that \( S \) is a set consisting of \( n \) (\( n \geq 3 \)) positive numbers. If three different elements in \( S \) can form the sides of a triangle, then \( S \) is called a "triangular set." Consider the set of consecutive positive integers \(\{4,5, \cdots, m\}\). All of its 10-element subsets are triangular sets. What is the maximum possible value of \( m \)?

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