
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
Two numbers \( x \) and \( y \) satisfy the equation \( 26x^2 + 23xy - 3y^2 - 19 = 0 \) and are respectively the sixth and eleventh terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression.

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