
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 $\overrightarrow{a} = (\cos\alpha, 1+\sin\alpha)$ and $\overrightarrow{b} = (1+\cos\alpha, \sin\alpha)$,
(1) If the magnitude of $\overrightarrow{a} + \overrightarrow{b}$ is $\sqrt{3}$, find the value of $\sin2\alpha$;
(2) Let vector $\overrightarrow{c} = (-\cos\alpha, -2)$, determine the range of values for the dot product $(\overrightarrow{a} + \overrightarrow{c}) \cdot \overrightarrow{b}$.

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