
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
For functions $y=f(x)$ and $y=g(x)$, defined on domains $D_{1}$ and $D_{2}$ respectively, define the function $h(x)$ as follows:
$$
h(x)= \begin{cases} 
f(x)\cdot g(x), & x\in D_{1} \text{ and } x\in D_{2} \\
f(x), & x\in D_{1} \text{ and } x\notin D_{2} \\
g(x), & x\notin D_{1} \text{ and } x\in D_{2}
\end{cases}
$$
Given $f(x)=x-2 (x\geqslant 1)$ and $g(x)=-2x+3 (x\leqslant 2)$, find the interval where $h(x)$ is monotonically decreasing.

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