
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
For the function $y=f\left(x\right)$, $x\in \left(0,+\infty \right)$, for any $a$, $b$, $c\in R$ and $a\geqslant 0$, $b\geqslant 0$, $c\geqslant 0$, if $f\left(a\right)$, $f\left(b\right)$, $f\left(c\right)$ are the lengths of the sides of a triangle, then the function $y=f\left(x\right)$ is called a "perfect triangle function" on $\left(0,+\infty \right)$.  
$(1)$ Given $\overrightarrow{a}=(\sqrt{3}\sin x,k\cos x)$, $\overrightarrow{b}=(2k\cos x,2\cos x)$, if the function $g(x)=\overrightarrow{a}\cdot\overrightarrow{b}-k+1$ is a "perfect triangle function" on $[0,\frac{\pi}{2}]$, find the range of real number $k$;  
$(2)$ Under the conditions of $(1)$ and $k \gt 0$, let $h(x)=\sin 2x-\frac{13\sqrt{2}}{5}\sin\left(x+\frac{\pi}{4}\right)+\frac{369}{100}$. If for any $x_{1}\in[0,\frac{\pi}{2}]$, there always exists $x_{2}\in[0,\frac{\pi}{2}]$ such that $g(x_{2})\geq h(x_{1})$, find the range of real number $k$.

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