
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 $\{a_{n}\}$ where $a_{1}=1$, $a_{2}=4$, and $2{a_n}=\frac{{n-1}}{n}{a_{n-1}}+\frac{{n+1}}{n}{a_{n+1}}$ for $n≥2$, find the maximum value of $\frac{{{a_n}}}{n}$.

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