
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 $f(x)=\lg (x+1)$.

$(i)$ If $0 < f(1-2x)-f(x) < 1$, find the range of real numbers $x$.

$(ii)$ If $g(x)$ is an even function with a period of $2$, and when $0 \leqslant x \leqslant 1$, $g(x)=f(x)$, and when $x \in [1,2]$, find the expression for the function $y=g(x)$.

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