
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 $\overrightarrow{a}, \overrightarrow{b}$ in a plane with an angle of ${120}^{^{\circ }}$ between them, and $\left|\overrightarrow{a}\right|=4$, $\left|\overrightarrow{b}\right|=2$.

(I) Find $(\overrightarrow{a}-2\overrightarrow{b})\cdot (\overrightarrow{a}+\overrightarrow{b})$

(II) Find $\left|3\overrightarrow{a}-4\overrightarrow{b}\right|$

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