
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]


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For a real-valued function  $f(x,y)$  of two positive real variables  $x$  and  $y$ , define  $f$  to be *linearly bounded* if and only if there exists a positive number  $K$  such that  $|f(x,y)| < K(x+y)$  for all positive  $x$  and  $y.$  Find necessary and sufficient conditions on the real numbers  $\alpha$  and  $\beta$  such that  $x^{\alpha}y^{\beta}$  is linearly bounded.

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