
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
In a certain basis of the three-dimensional linear space \(X_3\), the mappings are defined as follows:

$$
\begin{aligned}
x & \mapsto \widehat{A} x=\left\{x_{1}+x_{2}-x_{3}, x_{2}+x_{3}, x_{3}\right\} \\
x & \mapsto \widehat{B} x=\left\{x_{2}+2 x_{3},-x_{1}, x_{2}\right\}
\end{aligned}
$$

where \(x=\{x_{1}, x_{2}, x_{3}\}\) is an arbitrary vector in the space \(X_{3}\). Find the coordinates of the vector \((2 \widehat{A}+\widehat{A} \circ \widehat{B}) x\) in the same basis.

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