
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 a sequence $\{a_n\}$ that satisfies $a_1 = \frac{1}{2}$, and $a_{n+1} = \frac{na_n}{(n+1)(na_n + 1)}$ (where $n\in\mathbb{N}^*$), if the inequality $\frac{4}{n^2} + \frac{1}{n} + ta_n \geq 0$ always holds, then the range of values for the real number $t$ is __________.

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