
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 an even function defined on $\mathbb{R}$, such that for any $x \in \mathbb{R}$, $f(x) = f(x+4)$. Also, when $x \in [-2, 0]$, $f(x) = \left(\frac{1}{2}\right)^x - 1$. If the equation $f(x) - \log_a(x+2) = 0$ ($a > 1$) has exactly three distinct real roots in the interval $(-2, 6]$, then the range of values for $a$ is $\boxed{\text{answer}}$.

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