
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
Define an even function $f(x)$ on $\mathbb{R}$, such that for any $x \in \mathbb{R}$, $f(x+2)=f(x)$ holds, and when $x \in [-1,0]$, $f(x)=( \frac {1}{2})^{x}-1$. If the equation $f(x)-\log _{a}(x+1)=0$ ($a > 0$) has exactly $3$ distinct real roots in the interval $(-1,3]$, then the range of the real number $a$ is \_\_\_\_\_\_.

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