
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
Given a sequence $\{a_n\}$ satisfies $a_1=1$, $a_na_{n+1}=2^n$, $n\in\mathbb{N}$.
- $(1)$ If the function $f(x)=A\sin (2x+\phi)$ ($A > 0$, $0 < \phi < \pi$) reaches its maximum value $a_4+1$ at $x= \frac {\pi}{6}$, find the range of $f(x)$ in the interval $\left[- \frac {\pi}{12}, \frac {\pi}{2}\right]$.
- $(2)$ Find the general formula for the sequence $\{a_n\}$.

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