
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
Solve for integer values of \( x \) and \( y \) in the given system of equations:

\[ 
\begin{gathered}
4 x^{2} = y^{2} + 2 y + 1 + 3 \\
(2 x)^{2} - (y + 1)^{2} = 3 \\
(2 x - y - 1) \cdot (2 x + y + 1) = 3
\end{gathered}
\]

Given the integer solutions \( x \) and \( y \), we get the following systems to solve:

1. \( \left\{\begin{array}{l}2 x - y - 1 = 1 \\ 2 x + y + 1 = 3\end{array}\right. \)
2. \( \left\{\begin{array}{l}2 x - y - 1 = 3 \\ 2 x + y + 1 = 1\end{array}\right. \)
3. \( \left\{\begin{array}{l}2 x - y - 1 = -3 \\ 2 x + y + 1 = -1\end{array}\right. \)
4. \( \left\{\begin{array}{l}2 x - y - 1 = -1 \\ 2 x + y + 1 = -3\end{array}\right. \)

Solve each system to find solutions for \( x \) and \( y \).

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