
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 $\overrightarrow{c}$ lie in the same plane, and $\overrightarrow{a}=(3,2)$, $\overrightarrow{b}=(-2,1)$.
$(1)$ If $k\overrightarrow{a}-\overrightarrow{b}$ is perpendicular to $\overrightarrow{a}+2\overrightarrow{b}$, find the value of $k$.
$(2)$ If $\overrightarrow{c}=\overrightarrow{a}+x\overrightarrow{b}$ (where $x\in \mathbb{R}$), find the angle between vectors $\overrightarrow{c}$ and $\overrightarrow{b}$ when $|\overrightarrow{c}|$ is minimized.

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