
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 $f(x)$ that satisfies $f(2+x) \cdot f(2-x) = 4$. When $x \in [0,2]$, $f(x) = x^2 - ax + 2a - 2$ where $a > 0$.
$(1)$ If $f(2) + f(3) = 6$, find the value of $a$.
$(2)$ When $x \in [0,4]$, $1 \leq f(x) \leq 3$, find the range of values for $a$.

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