
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
Suppose the unique zero of the function $f(x)$ is simultaneously located in the intervals $(0,16)$, $(0,8)$, $(0,4)$, and $(0,2)$. Among the following statements, the correct one is $\_\_\_\_\_\_$ (fill in the serial number).
$①$ The function $f(x)$ has a zero in the interval $(0,1)$;
$②$ The function $f(x)$ has a zero in either the interval $(0,1)$ or $(1,2)$;
$③$ The function $f(x)$ has no zeros in the interval $[2,16)$;
$④$ The function $f(x)$ has no zeros in the interval $(1,16)$.

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