
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
For positive integer n, define  $a_n$  as the number of the triangles with integer length of every side and the length of the longest side being  $2n.$ (1) Find  $a_n$  in terms of  $n;$ (2)If the sequence  $\{ b_n\}$  satisfying for any positive integer  $n,$   $\sum_{k=1}^n(-1)^{n-k}\binom {n}{k} b_k=a_n.$  Find the number of positive integer  $n$  satisfying that  $b_n\leq 2019a_n.$ 

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