
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 all $x\in R$, the domain of functions $f(x)$ and $g(x)$ is $R$, $f(x+\frac{5}{2})+f(x)=2$, and $y=f(1+2x)$ is an even function. Also, $g(x+2)-g(x-2)=0$, and $y=g(x+1)-1$ is an odd function. For all $x\in [0,2]$, $f(x)+g(x)=3^{x}+x^{3}$. Find $f(2022)\cdot g(2022)$.

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