
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
Let \( M(x, y, z) \) be the minimum of the three numbers \( x, y, z \). If the quadratic function \( f(x) = ax^2 + bx + c \) (where \( a, b, c > 0 \)) has roots, then find the maximum value of \( M\left( \frac{b+c}{a}, \frac{c+a}{b}, \frac{a+b}{c} \right) \).

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