
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 $4 \times 4$ matrix $\mathbf{M}$ such that
\[\mathbf{M} \mathbf{u} = -2 \mathbf{u}\] for all four-dimensional vectors $\mathbf{u}$, and
\[\mathbf{M} \mathbf{e_4} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ -2 \end{pmatrix},\]
where $\mathbf{e_4} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}$ is the fourth standard basis vector.

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