
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 two vectors $\vec{a}$ and $\vec{b}$ on a plane with an angle of ${60}^{\circ }$ between them, $\vec{a}=(2,0)$, and $|\vec{b}|=1$, find the value of $|\vec{a}+2\vec{b}|$.

A) $6$
B) $36$
C) $2\sqrt{3}$
D) $12$

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