
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 \( k \) be a real number. For any real number \( x \), define \( f(x) = \frac{x^{4} + kx^{2} + 1}{x^{4} + x^{2} + 1} \).

(1) Find the maximum and minimum values of \( f(x) \).

(2) Determine all real numbers \( k \) such that, for any three real numbers \( a, b, \) and \( c \), there exists a triangle with side lengths \( f(a) \), \( f(b) \), and \( f(c) \).

(1994 Bulgarian Mathematical Olympiad)

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