
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 polar equations of curves $E_1$ and $E_2$ are $\rho=4\cos\theta$ and $\rho\cdot\cos\left(\theta- \frac{\pi}{4}\right)=4$, respectively. Rotate curve $E_1$ counterclockwise around the pole by an angle $\alpha$, where $\alpha\in\left(0, \frac{\pi}{2}\right)$, to obtain curve $E_3$.  
(1) When $\alpha= \frac{\pi}{6}$, find the polar equation of curve $E_3$;  
(2) When $E_3$ and $E_2$ have exactly one common point, find $\alpha$.

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