
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 an operation: $({a,b})[{\begin{array}{l}c\\ d\end{array}}]=ac+bd$.  
$(1)$ Given $z$ is a complex number, and $({3,\overline{z}})[{\begin{array}{l}z\\ 4\end{array}}]=7-3i$, find $|z|$;  
$(2)$ Given $x$ and $y$ are real numbers, $({y+\sin2x,2})[{\begin{array}{l}i\\ y\end{array}}]-({1,{{\sin}^2}x})[{\begin{array}{l}{\sin x}\\{2\sqrt{3}i}\end{array}}]$ is also a real number. Express $y$ as a function of $x$ and find the interval where this function is monotonically increasing.

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