
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
Given the matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the operation $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}$ transforms the point $(x, y)$ to the point $(ax + by, cx + dy)$ on the plane.
(1) Find the coordinates of the point formed by the action of the matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ on the point $(2, 3)$.
(2) If the curve $x^2 + 4xy + 2y^2 = 1$ is transformed into the curve $x^2 - 2y^2 = 1$ by the matrix $\begin{pmatrix} 1 & a \\ b & 1 \end{pmatrix}$, find the value of $a + b$.

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