
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
(1) Given $\vec{a} = (1, 0)$, $\vec{b} = (-1, 1)$, and $\vec{c} = 2\vec{a} + \vec{b}$. Find the magnitude of $\vec{c}$.
(2) If $|\vec{a}| = 2$, $|\vec{b}| = 1$, and the angle between $\vec{a}$ and $\vec{b}$ is $60^\circ$, calculate $\vec{a} \cdot (\vec{a} + \vec{b})$.

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