
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 $\overrightarrow{a}=(1,2)$, $\overrightarrow{b}=(3,1)$, $\overrightarrow{c}= \overrightarrow{b}-k \overrightarrow{a}$, and $\overrightarrow{a} \perp \overrightarrow{c}$.  
$(1)$ Find the projection of vector $\overrightarrow{b}$ in the direction of vector $\overrightarrow{a}$.  
$(2)$ Find the value of the real number $k$ and the coordinates of vector $\overrightarrow{c}$.

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