
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 any positive integer \( n \), let \( a_{n} \) denote the number of triangles with integer side lengths whose longest side is \( 2n \).

(1) Find an expression for \( a_{n} \) in terms of \( n \);

(2) Given the sequence \( \{b_{n}\} \) satisfies
$$
\sum_{k=1}^{n}(-1)^{n-k} \binom{n}{k} b_{k}=a_{n}\quad (n \in \mathbf{Z}_{+}),
$$
find the number of positive integers \( n \) such that \( b_{n} \leq 2019 a_{n} \).

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