
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
(1) Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ have an angle of $120^{\circ}$ between them, and $|\overrightarrow{a}|=4$, $|\overrightarrow{b}|=2$, find $|\overrightarrow{a}+ \overrightarrow{b}|$;  
(2) Given $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=3$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, $\overrightarrow{c}=5\overrightarrow{a}+3\overrightarrow{b}$, $\overrightarrow{d}=3\overrightarrow{a}+k\overrightarrow{b}$, for what value of the real number $k$ will $\overrightarrow{c} \perp \overrightarrow{d}$.

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