
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 inequality $k4^{x} - 2^{x+1} + 6k < 0$ with respect to $x$,

(1) If the solution set of the inequality is $(1, \log_{2}3)$, find the value of the real number $k$;

(2) If the inequality holds for all $x \in (1, \log_{2}3)$, find the range of values for the real number $k$;

(3) If the solution set of the inequality is a subset of $(1, \log_{2}3)$, find the range of values for the real number $k$.

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