
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 $\overset{⇀}{a} =3 \overset{⇀}{{e}\_{1}} -3 \overset{⇀}{{e}\_{2}}$ and $\overset{⇀}{b} =4 \overset{⇀}{{e}\_{1}} + \overset{⇀}{{e}\_{2}}$, where $\overset{⇀}{{e}\_{1}} =(1,0)$ and $\overset{⇀}{{e}\_{2}} =(0,1)$, find:

1. The dot product $\overset{⇀}{a} \cdot \overset{⇀}{b}$ and the magnitude $|\overset{⇀}{a} + \overset{⇀}{b}|$.
2. The cosine value of the angle between $\overset{⇀}{a}$ and $\overset{⇀}{b}$.

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