
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
Let $\overrightarrow{a}=(-1,1)$, $\overrightarrow{b}=(x,3)$, $\overrightarrow{c}=(5,y)$, $\overrightarrow{d}=(8,6)$, and $\overrightarrow{b}// \overrightarrow{d}$, $(4 \overrightarrow{a}+ \overrightarrow{d})\perp \overrightarrow{c}$.  
$(1)$ Find $\overrightarrow{b}$ and $\overrightarrow{c}$;  
$(2)$ Find the projection of $\overrightarrow{c}$ in the direction of $\overrightarrow{a}$.

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