
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 $$\overrightarrow {a} = (1+\cos\alpha, \sin\alpha)$$, $$\overrightarrow {b} = (1-\cos\beta, \sin\beta)$$, $$\overrightarrow {c}=(1,0)$$, with $\alpha \in (0,\pi)$, $\beta \in (\pi,2\pi)$, the angle between vector $$\overrightarrow {a}$$ and $$\overrightarrow {c}$$ is $\theta_1$, the angle between vector $$\overrightarrow {b}$$ and $$\overrightarrow {c}$$ is $\theta_2$, and $\theta_1 - \theta_2 = \frac {\pi}{6}$. If in triangle ABC, the sides opposite angles A, B, and C are a, b, and c respectively, and angle A = $\beta - \alpha$.

Find:  
(Ⅰ) The magnitude of angle A;  
(Ⅱ) If the circumradius of triangle ABC is $4\sqrt{3}$, determine the range of values for b+c.

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