
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  $f(x,\ y)=\frac{x+y}{(x^2+1)(y^2+1)}.$ (1) Find the maximum value of  $f(x,\ y)$  for  $0\leq x\leq 1,\ 0\leq y\leq 1.$ (2)  Find the maximum value of  $f(x,\ y),\ \forall{x,\ y}\in{\mathbb{R}}.$ 

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