
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

Consider all non-negative numbers \(a\) and \(b\) such that \(0 \leqq a < b\), and for all \(x\), the function

\[ P(x) = a + b \cos x + \frac{b}{2 \sqrt{2}} \cos 2x \]

is non-negative.

Determine the set of possible values for the ratio \(\frac{b + a}{b - a}\).

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