
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
Find the matrix of the operator $\widehat{A}$ in the basis $\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \mathbf{e}_{3}^{\prime}$, where

$$
\begin{aligned}
& \mathbf{e}_{1}^{\prime}=\mathbf{e}_{1}+\mathbf{e}_{2}+2 \mathbf{e}_{3}, \\
& \mathbf{e}_{2}^{\prime}=2 \mathbf{e}_{1}-\mathbf{e}_{2} \\
& \mathbf{e}_{3}^{\prime}=-\mathbf{e}_{1}+\mathbf{e}_{2}+\mathbf{e}_{3},
\end{aligned}
$$

if in the basis $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ its matrix is given by

$$
A_{\mathbf{e}}=\left(\begin{array}{rrr}
2 & 0 & -1 \\
0 & 1 & -2 \\
-1 & 2 & 0
\end{array}\right)
$$

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