
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 $\overrightarrow {a} = (\cos\alpha, \sin\alpha)$, $\overrightarrow {b} = (\cos x, \sin x)$, $\overrightarrow {c} = (\sin x + 2\sin\alpha, \cos x + 2\cos\alpha)$, where $0 < \alpha < x < \pi$.  
(1) If $\alpha= \frac {\pi}{4}$, find the minimum value of the function $f(x) = \overrightarrow {b} \cdot \overrightarrow {c}$ and the corresponding value of $x$;  
(2) If the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$ is $\frac {\pi}{3}$, and $\overrightarrow {a}$ is perpendicular to $\overrightarrow {c}$, find the value of $\tan 2\alpha$.

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