
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
Reading material: Jun, who is good at thinking, used a "whole substitution" method to solve the system of equations $\left\{\begin{array}{l}{2x+5y=3①}\\{4x+11y=5②}\end{array}\right.$:<br/>Solution: Transform equation ②: $4x+10y+y=5$, that is $2\left(2x+5y\right)+y=5$③.<br/>Substitute equation ① into ③: $2\times 3+y=5$, therefore $y=-1$,<br/>So, substituting $y=-1$ into ① gives $x=4$, therefore the solution to the system of equations is $\left\{\begin{array}{l}{x=4}\\{y=-1}\end{array}\right.$.<br/>Please solve the following problems:<br/>$(1)$ Imitate Jun's "whole substitution" method to solve the system of equations $\left\{\begin{array}{l}{3x-2y=5①}\\{9x-4y=19②}\end{array}\right.$;<br/>$(2)$ Given that $x$ and $y$ satisfy the system of equations $\left\{\begin{array}{l}{3x^{2}-2xy+12y^{2}=47①}\\{2x^{2}+xy+8y^{2}=36②}\end{array}\right.$, find the value of $x^{2}+4y^{2}$.

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