
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 three vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ that satisfy: $\overrightarrow{a}=(2\sqrt{3}\sin {60}^{\circ },4)$, $\overrightarrow{b}=(2,6\sqrt{2}\cos \frac{3\pi }{4})$, $\overrightarrow{c}=(4,{1})$.

(1) Find the value of $\overrightarrow{a}\cdot (\overrightarrow{a}+\overrightarrow{b})$;

(2) Represent $\overrightarrow{c}$ using $\overrightarrow{a}$ and $\overrightarrow{b}$ as the basis.

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