
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 the vectors $\overrightarrow{a} = (\lambda+2, \lambda^2 - \sqrt{3}\cos\alpha)$ and $\overrightarrow{b} = (m, \frac{m+\sin\alpha}{2})$, where $\lambda$, $m$, and $\alpha$ are real numbers.  
(1) If $\lambda=m=0$, and $\overrightarrow{a} \cdot \overrightarrow{b} = \cos2\alpha + \frac{1}{8}$, find $\tan\alpha$.  
(2) 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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