
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 integer solutions to the equation: \[2x^4 - 4y^4 - 7x^2y^2 - 27x^2 + 63y^2 + 85 = 0\]

Translate the equation using the substitutions \(x^2 = a\) and \(y^2 = b\), resulting in the quadratic equation: \[2a^2 - 4b^2 - 7ab - 27a + 63b + 85 = 0\]

Solve for the possible values of \(a\) and \(b\) that lead to integer solutions for \(x\) and \(y\).

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