
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 the domain of the function $f(x)$ is $(0,+\infty)$, and it satisfies $f(xy)=f(x)+f(y)$, with $f(2)=1$. If $f(x) < f(y)$ for $0 < x < y$, find the solution set for the inequality $f(x-1)+f(x+1) < 2$ (expressed as a set).

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