
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 about $x$, $x^{2}+n^{2}-x < 3nx-n^{2}-n (n \in \mathbb{N}^{*})$, let $a_{n}$ denote the number of integers in the solution set. Let $D_{n}$ denote the sum of the first $n$ terms of the sequence $\{\frac{2a_{n}+1}{2^{n}}\}$. Find the smallest integer $t$ such that $D_{n} < t$ holds true for all $n \in \mathbb{N}^{*}$.

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