
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
There exist vectors $\mathbf{a}$ and $\mathbf{b}$ such that
\[\mathbf{a} + \mathbf{b} = \begin{pmatrix} 10 \\ -5 \\ 0 \end{pmatrix},\]
where $\mathbf{a}$ is parallel to $\begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix},$
and $\mathbf{b}$ is orthogonal to $\begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}.$
Find $\mathbf{b}.$

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