
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 scale factor \(k > 0\), and let \(\mathbf{R}\) be a matrix representing a rotation about the origin by an angle of \(\theta\) counter-clockwise. Following the rotation and dilation, a translation by vector \(\mathbf{v} = \begin{pmatrix} 3 \\ 3 \end{pmatrix}\) is applied, resulting in 
\[
\mathbf{R} \mathbf{D} + \mathbf{v} = \begin{pmatrix} 11 & -5 \\ 7 & 11 \end{pmatrix}.
\]
Find the value of \(\tan \theta\).

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