
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 $f(x)=2\cos\omega x+2\sqrt{3}\cos\left(\omega x+\frac{3\pi}{2}\right)$, where $\omega \in \mathbb{N}^{*}$. If for $\forall \lambda \in \mathbb{R}$, there exist at least two distinct real numbers $m$ and $n$ in the interval $[\lambda, \lambda +3]$ such that $f(m)f(n)=16$, determine the minimum value of $\omega$.

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