
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 $\overrightarrow{a}=({1,2})$ and $\overrightarrow{b}=({3,-2})$.
$(Ⅰ)$ Find $|{\overrightarrow{a}-\overrightarrow{b}}|$.
$(Ⅱ)$ Find the cosine value of the angle $\theta$ between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.
$(Ⅲ)$ If $|{\overrightarrow{c}}|=\sqrt{10}$ and $({2\overrightarrow{a}+\overrightarrow{c}})⊥\overrightarrow{c}$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{c}$.

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