
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 $f(x) = \log_{a}4 \cdot \log_{2}(2+x) + 2\log_{a}(4-x)$, where $a > 0$ and $a \neq 1$.
$(1)$ If $f(2) = 6$, find the value of the real number $a$ and the domain of the function $f(x)$.
$(2)$ Find the range of the function $f(x)$.

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