
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$  and  $ b$  be positive real numbers with  $ a\ge b$ . Let  $ \rho$  be the maximum possible value of  $ \frac{a}{b}$  for which the system of equations
\[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in  $ (x,y)$  satisfying  $ 0\le x<a$  and  $ 0\le y<b$ . Then  $ \rho^2$  can be expressed as a fraction  $ \frac{m}{n}$ , where  $ m$  and  $ n$  are relatively prime positive integers. Find  $ m\plus{}n$ .

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