
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
The equation \( x^2 + y^2 = z^2 \) satisfies \( \gcd(x, y) = 1 \) and \( 2 \mid y \). All positive integer solutions \( (x, y, z) \) can be represented as \( x = a^2 - b^2 \), \( y = 2ab \), \( z = a^2 + b^2 \), where \( a \) and \( b \) are any integers satisfying \( a > b > 0 \), with \( a \) and \( b \) being one odd and one even, and \( \gcd(a, b) = 1 \).

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