
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
Determine all possible values of the parameter \(a\) such that for all \(x\) and \(y\), the inequality \(2a x^{2} + 2a y^{2} + 4a x y - 2 x y - y^{2} - 2 x + 1 \geq 0\) holds.

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