
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
For any  $n,T \geq 2, n, T \in \mathbb{N}$ , find all  $a \in \mathbb{N}$  such that  $\forall a_i > 0, i = 1, 2, \ldots, n$ , we have

\[\sum^n_{k=1} \frac{a \cdot k + \frac{a^2}{4}}{S_k} < T^2 \cdot \sum^n_{k=1} \frac{1}{a_k},\] where  $S_k = \sum^k_{i=1} a_i.$ 

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