
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) Find the range of real number $k$ such that for any $x \in \mathbb{R}$, $f(x) > 0$ always holds;
(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 always exists a triangle with side lengths $f(x_1)$, $f(x_2)$, and $f(x_3)$, find the range of real number $k$.

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