
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 $\{a_{n}\}(n$ is a positive integer) with a recursive formula $\left\{\begin{array}{l}a_{n}=3a_{n-1}+4\left({n≥2}\right)\;,\;\\ a_{1}=1.\end{array}\right.$ If there exists a positive integer $n$ such that $n(2n+1)\geqslant t(a_{n}+2)$, then the maximum value of $t$ is ____.

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