
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 a function $f(x)$ defined on $\mathbb{R}$, for any $x_1, x_2$ ($x_1 \neq x_2$), it holds that $(x_1-x_2)[f(x_1)-f(x_2)] < 0$, and the graph of the function $y=f(x-1)$ is symmetric about the point $(1,0)$. If $f(1)=-1$, then determine the range of $x$ that satisfies $-1 \leqslant f\left(\frac{1}{2}x-1\right) \leqslant 1$.

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