
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 plane vectors $\overrightarrow{a}=(\sin x,-\cos x)$, $\overrightarrow{b}=(\sqrt{3}\cos x,-\cos x)$, $\overrightarrow{c}=(1,2\cos x-1)$.
$(1)$ Let $f(x)=2\overrightarrow{a}\cdot \overrightarrow{b}$. Find the smallest positive period of $f(x)$.
$(2)$ Let $y=f(x)+\overrightarrow{a}\cdot \overrightarrow{c}$.
① Let $t=\sin x+\cos x$. Express $\sin x\cdot \cos x$ in terms of $t$ and write down the range of $t$.
② Find the minimum value of $y$.

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