
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 that $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ are three vectors in the same plane, where $\overrightarrow{a}=(3,4)$.  
$(1)$ If $|\overrightarrow{c}|=10$ and $\overrightarrow{c} \parallel \overrightarrow{a}$, find the coordinates of $\overrightarrow{c}$;  
$(2)$ If $|\overrightarrow{b}|=\sqrt{10}$ and $\overrightarrow{a}+2\overrightarrow{b}$ is perpendicular to $2\overrightarrow{a}-\overrightarrow{b}$, find the projection vector of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$.

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