
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 by $f(x+2)=2f(x)$ and $$f(x) = \begin{cases} -x^2+4x, & 2 \leq x \leq 3 \\ \frac{x^2+2}{x}, & 3 < x < 4 \end{cases},$$
determine the range of values for the real number $a$ such that for any $x_1 \in [-2,0)$, there exists an $x_2 \in [-2,1]$ such that $g(x_2)=f(x_1)$, where $g(x)=ax+1$.

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