
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 that $f(x)=\log(x+1)$,  
(1) If $0<f(1-2x)-f(x)<1$, determine the range of x;  
(2) If $g(x)$ is an even function with a period of 2 and $g(x)=f(x)$ when $0\leq x\leq 1$, find the inverse function of $y=g(x)$ for $x\in[1,2]$.

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