
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 vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2,|\overrightarrow{b}|=1$, the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60°$. Let $\overrightarrow{c}=2\overrightarrow{a}+3\overrightarrow{b}$ and $\overrightarrow{d}=3\overrightarrow{a}+k\overrightarrow{b}$. Find the value of the real number $k$ when:<br/>
$(1)$ $\overrightarrow{c}⊥\overrightarrow{d}$;<br/>
$(2)$ $|\overrightarrow{d}|=2\sqrt{13}$.

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