
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 $f(x) = 2^x$, its inverse function is $f^{-1}(x)$  
(1) If $f^{-1}(x) - f^{-1}(1-x) = 1$, find the value of the real number $x$;  
(2) If the equation about $x$, $f(x) + f(1-x) - m = 0$, has a solution in the interval $[0, 2]$, find the range of the real number $m$.

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