
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
Jiajia and Qiqi, two classmates, solved the equation $3\left(x-3\right)=\left(x-3\right)^{2}$ as follows:

Jiajia:  
Divide both sides by $\left(x-3\right)$, we get  
$3=x-3$,  
thus $x=6$.

Qiqi:  
Rearrange the equation, we get $3\left(x-3\right)-\left(x-3\right)^{2}=0$,  
Factor out the common factor, we get $\left(x-3\right)\left(3-x-3\right)=0$.  
Thus, $x-3=0$ or $3-x-3=0$,  
solving which we get $x_{1}=3$, $x_{2}=0$.

$(1)$ Jiajia's method is ______; Qiqi's method is ______; (Fill in "correct" or "incorrect")  
$(2)$ Please choose an appropriate method to try to solve the quadratic equation $4x\left(2x+1\right)=3\left(2x+1\right)$.

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