
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 $f(x) = 2x \ln(2x - 1) - \ln(2x - 1) - mx + \frac{1}{e}$.
(1) Find the range of real number $m$ if the equation $f(x) = 0$ has real roots in the interval $(\frac{3}{5}, \frac{e^2 + 1}{2})$.
(2) Find the value of real number $m$ if the minimum value of $y = f(x)$ in the interval $[1, \frac{3}{2}]$ is $-4 + \frac{1}{e}$.

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