
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 the function $y=2\sin (\omega x+ \varphi )(\omega > 0,|\varphi | < \dfrac{{ }\, \pi\, { }}{2})$.
$①$ If $f(0)=1$, then $\varphi =$ \_\_\_;
$②$ If $\exists x\in \mathbf{R}$, such that $f(x+2)-f(x)=4$ holds true, then the minimum value of $\omega$ is \_\_\_.

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