
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
How many decreasing sequences  $a_1, a_2, \ldots, a_{2019}$  of positive integers are there such that  $a_1\le 2019^2$  and  $a_n + n$  is even for each  $1 \le n \le 2019$ ?

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