
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 $x \in \mathbb{R}$, $f(x+4)=f(x)$, and when $x \in [-2,0]$, $f(x)=2-\left(\frac{1}{2}\right)^x$. If the equation $f(x)-\log_a(x+2)=0\ (0 < a < 1)$ has exactly three distinct real roots in the interval $(-2,6]$, find the range of values for the real number $a$.

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