
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 $\overrightarrow{b}=(4,-3)$.  
$(Ⅰ)$ Find $\overrightarrow{a} \cdot \overrightarrow{b}$;  
$(Ⅱ)$ If vector $\overrightarrow{c}$ is parallel to $\overrightarrow{a}$ and $|\overrightarrow{c}|=2\sqrt{5}$, find the coordinates of vector $\overrightarrow{c}$;  
$(Ⅲ)$ If $\overrightarrow{b}+k\overrightarrow{a}$ and $\overrightarrow{b}-k\overrightarrow{a}$ are perpendicular to each other, find the value of the real number $k$.

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