
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 matrix $\mathbf{P}$ that triples the first column of a matrix and simultaneously rotates the second column by 90 degrees counterclockwise. Specifically, for any matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the transformation should result in $\begin{pmatrix} 3a & -d \\ 3c & b \end{pmatrix}$. If no such matrix $\mathbf{P}$ exists, then enter the zero matrix.

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