
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
The vector $\mathbf{x}$ in the basis $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ has coordinates $\{1,2,3\}$. Find the coordinates of the vector $\mathbf{x}$ in the basis $\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \mathbf{e}_{3}^{\prime}$, where

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

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