
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
Let $f(x)$ be a function defined on $\left(0,+\infty \right)$, satisfying $f(xy)=f(x)+f(y)$, $f(\frac{1}{2})=1$, and for any $0 \lt \alpha\ \ \lt \beta $, $f(\alpha) \gt f(\beta)$.  
$(Ⅰ)$ Find $f(1)$;  
$(Ⅱ)$ If $f(2x)-f(2-x)\geqslant -1$, find the range of real number $x$.

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