
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 $f(x) = \log_{2}(2^{x}-1)$,  
(Ⅰ) Find the intervals of monotonicity for the function $f(x)$;  
(Ⅱ) If the function $g(x) = \log_{2}(2^{x}+1)$, and the equation $g(x) = m + f(x)$ has a solution in the interval $[1, 2]$, find the range of the real number $m$.

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