
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  $\mathbb{N}$  denote the set of all positive integers.Function  $f:\mathbb{N}\cup{0}\rightarrow\mathbb{N}\cup{0}$  satisfies :for any two distinct positive integer  $a,b$ , we have   $$ f(a)+f(b)-f(a+b)=2019 $$  (1)Find  $f(0)$ (2)Let  $a_1,a_2,...,a_{100}$  be 100 positive integers (they are pairwise distinct), find  $f(a_1)+f(a_2)+...+f(a_{100})-f(a_1+a_2+...+a_{100})$ 

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