
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 $\overrightarrow{a}=(6,1)$, $\overrightarrow{b}=(-2,3)$, $\overrightarrow{c}=(2,2)$, $\overrightarrow{d}=(-3,k)$.  
$(1)$ Find $\overrightarrow{a}+2\overrightarrow{b}-\overrightarrow{c}$;  
$(2)$ If $(\overrightarrow{a}+2\overrightarrow{c})\parallel(\overrightarrow{c}+k\overrightarrow{b})$, find the value of real number $k$.  
$(3)$ If the angle between $\overrightarrow{a}$ and $\overrightarrow{d}$ is obtuse, find the range of real number $k$.

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