
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 $f(x) = \frac{4^x - k \cdot 2^{x+1} + 1}{4^x + 2^x + 1}$, if there exists a triangle with sides of length $f(x_1)$, $f(x_2)$, and $f(x_3)$ for any real numbers $(x_1, x_2, x_3)$, then the range of values for the real number $k$ is _______.

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