
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 \( a, b, c \) be non-negative real numbers. Determine the minimum and maximum values of:
\[ S = \sqrt{\frac{ab}{(b+c)(c+a)}} + \sqrt{\frac{bc}{(a+c)(b+a)}} + \sqrt{\frac{ca}{(b+c)(b+a)}} \]

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