
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
Xiao Ming and Xiao Hua, two classmates, solved the quadratic equation $3\left(x-5\right)=\left(x-5\right)^{2}$ as shown in the following box:<br/>

| Xiao Ming | Xiao Hua |
|-----------|----------|
| $3=x-5$. Therefore, $x=8$. | Moving terms, we get $3\left(x-5\right)-\left(x-5\right)^{2}=0$. Factoring out, we get $\left(x-5\right)\left(3-x-5\right)=0$. Therefore, $x-5=0$ or $3-x-5=0$. Solving, we get $x_{1}=5$ and $x_{2}=-2$. |

$(1)$ Do you think their solutions are correct? Write down your judgment directly.<br/>
Xiao Ming's solution: ______. Xiao Hua's solution: ______. (Fill in "correct" or "incorrect")<br/>
$(2)$ Please choose an appropriate method to solve the quadratic equation $3\left(2x+1\right)=\left(2x+1\right)^{2}$.

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