
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
Example 21 (1994 Bulgarian Mathematical Olympiad problem) $k$ is a real number. For any real number $x$, let
$$
f(x)=\frac{x^{4}+k x^{2}+1}{x^{4}+x^{2}+1}.
$$
(1) Find the maximum and minimum values of $f(x)$.
(2) Find all real numbers $k$ such that for any 3 real numbers $a, b$, and $c$, there exists a triangle with side lengths $f(a), f(b)$, and $f(c)$.

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