
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 vector $\overrightarrow{a}=(3,-1)$, $|\overrightarrow{b}|=\sqrt{5}$, $\overrightarrow{a}\cdot\overrightarrow{b}=-5$, $\overrightarrow{c}=x\overrightarrow{a}+(1-x)\overrightarrow{b}$.
(Ⅰ) If $\overrightarrow{a}\perp\overrightarrow{c}$, find the value of the real number $x$;
(Ⅱ) When $|\overrightarrow{c}|$ takes the minimum value, find the cosine of the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$.

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