
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 quadratic function 
$$
f(x)=a x^{2}+b x+c \ \ (a, b, c > 0)
$$

which has roots, find the maximum value of 
$$
\min \left\{\frac{b+c}{a}, \frac{c+a}{b}, \frac{a+b}{c}\right\}.
$$

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