
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
Positive real numbers \( x \) and \( y \) satisfy the following conditions: There exist \( a \in[0, x] \) and \( b \in[0, y] \) such that \( a^{2} + y^{2} = 2 \), \( b^{2} + x^{2} = 1 \), and \( ax + by = 1 \). What is the maximum value of \( x + y \)?

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