
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 $f(x) = 1 - \frac{2}{2^x + 1}$, $g(x) = 2\sin(2x - \frac{\pi}{6})$.

(1) If the function $g(x) = (2^x + 1) \cdot f(x) + k$ has a root, find the range of the real number $k$;  
(2) For any $x_1 \in (0, 1)$, there always exists $x_2 \in \left[-\frac{\pi}{4}, \frac{\pi}{6}\right]$ such that the inequality $f(x_1) - m \cdot 2^{x_1} > g(x_2)$ holds. Find the range of the real number $m$.

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