
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 $\overrightarrow{a}$ and $\overrightarrow{b}$ are in the same plane, where $\overrightarrow{a} =(1,-2)$ and $|\overrightarrow{b}| =2\sqrt{5}$.

(Ⅰ) If $\overrightarrow{a} \parallel \overrightarrow{b}$, find the coordinates of vector $\overrightarrow{b}$;

(Ⅱ) If $(2\overrightarrow{a} -3\overrightarrow{b}) \cdot (2\overrightarrow{a} +\overrightarrow{b}) =-20$, find the value of the angle $\theta$ between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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