
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 $f(x)=\frac{{4^x+k\cdot{2^x}+1}}{{4^x+2^x+1}}$.  
$(1)$ If $f(x) > 0$ holds for any $x \in \mathbb{R}$, find the range of real number $k$;  
$(2)$ If the minimum value of $f(x)$ is $-2$, find the value of real number $k$;  
$(3)$ If for any $x_{1}$, $x_{2}$, $x_{3} \in \mathbb{R}$, there exists a triangle with side lengths $f(x_{1})$, $f(x_{2})$, $f(x_{3})$, find the range of real number $k$.

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