
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 quadratic inequality $x^2 - 2mx + m + 2 < 0$ ($x \in \mathbb{R}$) with the solution set $M$:

1. Find the range of $m$ when $M$ is an empty set.
2. Under the condition of (1), find the maximum value of $f(m) = \frac{2^m}{4^m + 1}$.
3. When $M$ is not an empty set and $M \subseteq [1, 4]$, find the range of real number $a$.

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