
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
The function $f^{-1}(x)$ is the inverse function of $f(x)=\log _{2}\left(2^{x}+1\right)$.

(1) If the equation $f^{-1}(x)=m+f(x)$ has a solution in the interval $[1,2]$, find the range of values for the real number $m$.

(2) If the inequality $f^{-1}(x)>m+f(x)$ has a solution in the interval $[1,2]$, find the range of values for the real number $m$.

(3) If the inequality $f^{-1}(x)>m+f(x)$ holds for any $x \in[1,2]$, find the range of values for the real number $m$.

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