
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 function \( f(x) = \ln{x} \) with its domain as \( (m, +\infty) \) where \( M > 0 \), ensure that for any \( a, b, c \in (M, +\infty) \) forming the sides of a right-angled triangle, \( f(a), f(b), \) and \( f(c) \) also form the sides of a triangle. Find the minimum value of \( M \).

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