
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 the vectors $\overrightarrow{a}=(3,2)$, $\overrightarrow{b}=(-1,2)$, and $\overrightarrow{c}=(4,1)$ in the plane.
1. Find the value of the real number $k$ such that $(\overrightarrow{a}+k\overrightarrow{c}) \perp (2\overrightarrow{b}-\overrightarrow{a})$.
2. If vector $\overrightarrow{d}$ is parallel to vector $\overrightarrow{c}$ and $|\overrightarrow{d}|=\sqrt{34}$, find the coordinates of vector $\overrightarrow{d}$.

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