
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 that the magnitudes of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are 1 and 2 respectively,
(1) If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi}{3}$, find the magnitude of $\overrightarrow{a} + 2\overrightarrow{b}$;
(2) If $(2\overrightarrow{a} - \overrightarrow{b}) \cdot (3\overrightarrow{a} + \overrightarrow{b}) = 3$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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