
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 $k \cdot 4^{x} - 2^{x+1} + 6k < 0$ for variable $x$:
1. If the solution set of the inequality is $\left\{x \mid 1 < x < \log_{2} 3\right\}$, find the value of the real number $k$;
2. If the solution set of the inequality is a subset of $\left\{x \mid 1 < x < \log_{2} 3\right\}$, find the range of values of the real number $k$;
3. If the inequality holds for all $x \in \left\{x \mid 1 < x < \log_{2} 3\right\}$, find the range of values of the real number $k$.

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