
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
Define the function $f(x)=a\sin x+b\cos x$ with the "adjoint vector" $\overrightarrow{OM}=(a,b)$, and the "adjoint function" of the vector $\overrightarrow{OM}=(a,b)$ as $f(x)=a\sin x+b\cos x$.  
$(1)$ Write down the "adjoint vector" $\overrightarrow{OM}$ of the function $g(x)=\cos (x+\frac{π}{3})+\cos x$, and find $|\overrightarrow{OM}|$;  
$(2)$ Let the vector $\overrightarrow{ON}=(1,\sqrt{3})$ have an adjoint function $\varphi(x)$, if the inequality $\varphi(x)+k\varphi(x+\frac{π}{2}) \gt 0$ holds for all $x∈[0,\frac{11π}{12}]$, find the range of real number $k$.

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