
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 comprehension: $a$, $b$, $c$, $d$ are real numbers. We call the symbol $|{\begin{array}{l}a&b\\ c&d\end{array}}|$ a $2\times 2$ determinant, and define it as $|{\begin{array}{l}a&b\\ c&d\end{array}}|=a\times d-b\times c$. For example, $|{\begin{array}{l}3&2\\{-1}&{-2}\end{array}}|=3\times({-2})-2\times({-1})=-6+2=-4$. The solution of a system of two linear equations $\left\{{\begin{array}{l}{{a_1}x+{b_1}y={c_1}}\\{{a_2}x+{b_2}y={c_2}}\end{array}}\right.$ can be expressed using $2\times 2$ determinants as $\left\{{\begin{array}{l}{x=\frac{{{D_x}}}{D}}\\{y=\frac{{{D_y}}}{D}}\end{array}}\right.$; where $D=|{\begin{array}{l}{{a_1}}&{{b_1}}\\{{a_2}}&{{b_2}}\end{array}}|$, ${D_x}=|{\begin{array}{l}{{c_1}}&{{b_1}}\\{{c_2}}&{{b_2}}|$, and ${D_y}=|{\begin{array}{l}{{a_1}}&{{c_1}}\\{{a_2}}&{{c_2}}\end{array}}|$.  
Problem solving:  
$(1)$ Calculate the value of the $2\times 2$ determinant $|{\begin{array}{l}{-2}&3\\ 2&1\end{array}}|$, which is ______.  
$(2)$ Solve the system of two linear equations $\left\{{\begin{array}{l}{2x-y=1}\\{3x+2y=11}\end{array}}\right.$ using the determinant method.

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