
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 that $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, with $a_1=1$ and $2S_n=(n+1)a_n$, if the solution set of the inequality $a_n^2 - ta_n \leqslant 2t^2$ for positive integers $n$ contains two distinct integer solutions, then the range of the positive real number $t$ is __________.

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