
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 $\overrightarrow{a}=(\sqrt{3},k)$, $\overrightarrow{b}=(0,-1)$, $\overrightarrow{c}=(1,\sqrt{3})$.  
$(Ⅰ)$ If $\overrightarrow{a}\perp \overrightarrow{c}$, find the value of $k$.  
$(Ⅱ)$ When $k=1$, and $\overrightarrow{a}-\lambda \overrightarrow{b}$ is collinear with $\overrightarrow{c}$, find the value of $\lambda$.  
$(Ⅲ)$ If $|\overrightarrow{m}|=\sqrt{3}|\overrightarrow{b}|$, and the angle between $\overrightarrow{m}$ and $\overrightarrow{c}$ is $150^{\circ}$, find $|\overrightarrow{m}+2\overrightarrow{c}|$.

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