
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 $f(x)=\sin^2 \omega x + (2 \sqrt{3} \sin \omega x - \cos \omega x) \cos \omega x - \lambda$ whose graph is symmetric about the line $x=\pi$, where $\omega$ and $\lambda$ are constants, and $\omega \in \left( \frac{1}{2},1\right)$.

 (Ⅰ) Find the smallest positive period of the function $f(x)$;

 (Ⅱ) If there exists ${x}_0 \in \left[0, \frac{3\pi}{5}\right]$ such that $f({x}_0)=0$, find the range of $\lambda$.

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