
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 $\vec{a}=( \sqrt{3},1)$, $\vec{b}=(0,-2)$, $\vec{c}=(k,\sqrt{3})$

$(1)$ Find $\vec{a}+2\vec{b}$;

$(2)$ For what value of $k$ is $\vec{a}+\vec{b}$ perpendicular to $\vec{c}$.

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