
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 a function $f(x)$ be an even function defined on $\mathbb{R}$, such that for all $x \in \mathbb{R}$, $f(x+2)=f(x)$. Also, when $x \in [-1, 0]$, $f(x) = \left(\frac {1}{2}\right)^x - 1$. If, in the interval $(-1, 3]$, the equation $f(x) - \log_a (x+1) = 0$ ($a > 0$) has exactly three distinct real roots, then the range of possible values for the real number $a$ is ______.

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