
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
Consider the function $f(x) = x^2 - 4ax + 3a$, where $a \in \mathbb{R}$.
1. If the solution set of the inequality $f(x) < 0$ is $\{x | 1 < x < m\}$, find the values of the real numbers $a$ and $m$.
2. If the solution set of the inequality $f(x) > 0$ is the entire set of real numbers $\mathbb{R}$, and for any $x \in [0, 1]$ the inequality $a^{k+3} < a^{x^2-kx} < a^{k-3}$ always holds, find the range of the real number $k$.

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