
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
A $1 \times 3$ rectangle is called a tromino. Petya and Vasya independently divide a $20 \times 21$ board into trominoes. Then, they compare the resulting divisions, and Petya pays Vasya an amount in rubles equal to the number of trominoes that coincidentally are in the same positions in both of their divisions. What is the maximum amount of money that Vasya can guarantee himself regardless of Petya's actions?

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