
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
Teacher $D$ thought of two positive integers $x$ and $y$ (with $y \geq x \geq 1$). He tells $A$ the value of $x + y$ and $B$ the value of $x \times y$. Then, $A$ and $B$ have the following conversation (both $A$ and $B$ know that $y \geq x \geq 1$):
$B$ says: I don't know the value of $x + y$.
$A$ says: Here’s a hint, the value of $x + y$ does not exceed 20. Once you can determine $x + y$ from this hint, then I will also know $x \times y$.
$B$ says: Now I know the value of $x + y$.

Find the values of $x$ and $y$.

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