
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
Let $a = 2b$ where $a$ and $b$ are positive real numbers, find the maximum value of $\alpha = \frac{a}{b}$ such that the system of equations 
\[
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
\]
holds for some $x, y$ with $0 \leq x < a$ and $0 \leq y < b$.

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