
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, three vectors are given: $\overrightarrow{a} = (3, 2)$, $\overrightarrow{b} = (-1, 2)$, and $\overrightarrow{c} = (4, 1)$.

(1) If $(\overrightarrow{a} + k\overrightarrow{c})$ is perpendicular to $(2\overrightarrow{b} - \overrightarrow{a})$, find the real number $k$;

(2) If vector $\overrightarrow{d}$ satisfies $\overrightarrow{d} \parallel \overrightarrow{c}$, and $|\overrightarrow{d}| = \sqrt{34}$, find the vector $\overrightarrow{d}$.

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