
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 $f(x)=\begin{cases}-x^{2}+2x-\frac{5}{4}, & x\leqslant 1 \\ {\log }_{ \frac{1}{3}}x-\frac{1}{4}, & x > 1\end{cases}$, $g(x)=|A-2|\cdot\sin x (x\in\mathbb{R})$, find the range of the real number $A$ such that for any $x_{1}$, $x_{2}\in\mathbb{R}$, it holds that $f(x_{1})\leqslant g(x_{2})$.

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