
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 $f(x)= \begin{cases} \ln (x+1),x > 0 \\ -x^{2}+2x,x\leqslant 0\\end{cases}$,
$(1)$ Determine the monotonicity of $f(x)$ using the definition or derivative method;
$(2)$ Find the solution set of the inequality $f(2x-1) > f(2-x)$.

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