
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 two non-collinear vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with magnitudes $|\overrightarrow{a}|=2$ and $|\overrightarrow{b}|=1$, and two more vectors $\overrightarrow{c}=2\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{d}=\overrightarrow{a}+k\overrightarrow{b}$,

(I) Find the value of the real number $k$ if $\overrightarrow{c}$ is parallel to $\overrightarrow{d}$.

(II) If $k=-7$ and $\overrightarrow{c}$ is perpendicular to $\overrightarrow{d}$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.

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