
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  $x$  and  $y$  be distinct real numbers such that
\[ \sqrt{x^2+1}+\sqrt{y^2+1}=2021x+2021y. \]
Find, with proof, the value of
\[ \left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right). \]

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