
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
Given the inequality system $\begin{cases} x > 0 \
y > 0 \
y \leqslant -nx + 2n \end{cases}$ representing a plane region $D_{n}$, let $a_{n}$ denote the number of lattice points within $D_{n}$, where $n \in \mathbb{N}^*$.

(1) Find the general formula for the sequence $\{a_{n}\}$;

(2) Let $b_{n} = 2^{a_{n}} + (-1)^{n}a_{n}$, find the sum of the first $2n$ terms of the sequence $\{b_{n}\}$.

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