
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 $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ are three vectors in the same plane, where $\overrightarrow{a}=(1,\;2)$.  
(Ⅰ) If $|\overrightarrow{b}|=3\sqrt{5}$ and $\overrightarrow{b} \parallel \overrightarrow{a}$, find the coordinates of $\overrightarrow{b}$.  
(Ⅱ) If the cosine of the angle $\theta$ between $\overrightarrow{c}$ and $\overrightarrow{a}$ is $-\frac{\sqrt{5}}{10}$, and $(\overrightarrow{a}+ \overrightarrow{c}) \perp (\overrightarrow{a}-9\overrightarrow{c})$, find $|\overrightarrow{c}|$.

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