
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
\[ f(x) = \log_{2}(4^x + 1) + mx \quad (m > 0) \]
the equation regarding \( x \)
\[ f\left(8\left(\log_{4} x\right)^{2} + 2 \log_{2} \frac{1}{x} + \frac{4}{m} - 4\right) = 1 \]
has exactly two different real solutions in the interval \([1, 2\sqrt{2}]\). Then the set of all possible values of \( m \) is

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