
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 \( f(x) = \frac{x^{4} + k x^{2} + 1}{x^{4} + x^{2} + 1} \) where \( k \in \mathbb{R} \) and \( x \in \mathbb{R} \):
1. Find the maximum and minimum values of \( f(x) \).
2. Find all real numbers \( k \) such that for any three real numbers \( a, b, c \), there exists a triangle with side lengths \( f(a), f(b), f(c) \).

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