
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 $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=3$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, with $\overrightarrow{c}=5\overrightarrow{a}+3\overrightarrow{b}$ and $\overrightarrow{d}=3\overrightarrow{a}+k\overrightarrow{b}$, find the value of the real number $k$ when

(1) $\overrightarrow{c}$ is parallel to $\overrightarrow{d}$;

(2) $\overrightarrow{c}$ is perpendicular to $\overrightarrow{d}$.

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