
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
Nina and Tadashi play the following game. Initially, a triple  $(a, b, c)$  of nonnegative integers with  $a+b+c=2021$  is written on a blackboard. Nina and Tadashi then take moves in turn, with Nina first. A player making a move chooses a positive integer  $k$  and one of the three entries on the board; then the player increases the chosen entry by  $k$  and decreases the other two entries by  $k$ . A player loses if, on their turn, some entry on the board becomes negative.

Find the number of initial triples  $(a, b, c)$  for which Tadashi has a winning strategy.

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