
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 vectors $\overrightarrow{a}=(\lambda+2,\lambda^{2}-\sqrt{3}\cos 2\alpha)$ and $\overrightarrow{b}=(m, \frac{m}{2}+\sin \alpha\cos \alpha)$, where $\lambda$, $m$, and $\alpha$ are real numbers.
(Ⅰ) If $\alpha= \frac{\pi}{12}$ and $\overrightarrow{a} \perp \overrightarrow{b}$, find the range of values for $m$.
(Ⅱ) If $\overrightarrow{a}=2\overrightarrow{b}$, find the range of values for $\frac{\lambda}{m}$.

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