
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 $R$ such that $f(x+2)=2f(x)$. When $x\in (2,4]$, $f(x)=\left\{\begin{array}{c}-x^{2}+4x, 2≤x≤3\\ \frac{x^{2}+2}{x}, 3<x≤4\end{array}\right.$. Let $g(x)=ax+1$. For $\forall x_{1}\in (-4,-2]$, there exists $x_{2}\in [-2,1]$, such that $g(x_{2})=f(x_{1})$. The range of real number $a$ is ______.

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