
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
Let vector $\overrightarrow{a}=(\cos \omega x-\sin \omega x,-1)$, $\overrightarrow{b}=(2\sin \omega x,-1)$, where $\omega > 0$, $x\in R$. It is known that the function $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$ has a minimum positive period of $4\pi$.

(Ⅰ) Find the value of $\omega$;

(Ⅱ) If $\sin x_{0}$ is a root of the equation $2t^{2}-t-1=0$, and $x_{0} \in \left( -\frac{\pi }{2},\frac{\pi }{2} \right)$, find the value of $f(x_{0})$.

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