
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] First carefully read the material, then try to solve the problem. The formula for the square of the sum (difference) of two numbers, $x^{2}\pm 2xy+y^{2}=\left(x\pm y\right)^{2}$, and the fact that the value of $\left(x\pm y\right)^{2}$ is always non-negative, have wide applications in mathematical studies. For example, when exploring the minimum value of the polynomial $x^{2}+6x-4$, we can proceed as follows: 

Solution: The original expression $=x^{2}+6x-4=x^{2}+6x+9-9-4=\left(x+3\right)^{2}-13=\left(x+3\right)^{2}-13$. Since the value of $\left(x+3\right)^{2}$ is always non-negative regardless of the value of $x$, the minimum value of $\left(x+3\right)^{2}$ is $0$, which occurs when $x=-3$. Therefore, the minimum value of $\left(x+3\right)^{2}-13$ is $0-13=-13$. Thus, when $x=-3$, the minimum value of the original polynomial is $-13$.

Problem: Following the above solution approach, determine the maximum value of the polynomial $-3x^{2}-6x+12$ and write down the corresponding values of $x$.

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