
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
In the plane Cartesian coordinate system $xOy$, let the two unit vectors in the direction of the $x$-axis and $y$-axis be $\overrightarrow{i}$ and $\overrightarrow{j}$, respectively. Given $\overrightarrow{OA}=\overrightarrow{i}+2\overrightarrow{j}$ and $\overrightarrow{OB}=2\overrightarrow{i}-4\overrightarrow{j}$. <br/>$(1)$ Find the cosine value of the angle between vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$; <br/>$(2)$ If point $P$ is the midpoint of the line segment $AB$, and vector $\overrightarrow{OP}$ is perpendicular to $\overrightarrow{OA}+k\overrightarrow{OB}$, find the value of the real number $k$.

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