
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
Let $\mathbf{D}$ be a matrix representing a dilation with a scale factor $3 > 0$, and let $\mathbf{M}$ be a matrix representing a reflection about the x-axis followed by a rotation about the origin by an angle $\theta$ counter-clockwise. If 
\[\mathbf{M} \mathbf{D} = \begin{pmatrix} -9 & 3\sqrt{3} \\ 3\sqrt{3} & 9 \end{pmatrix},\]
find $\tan \theta$.

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