
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 the sequence $\{{a_n}\}$ is a monotonically increasing sequence, $S_n$ is the sum of its first $n$ terms, and $2S_n = a_n^2 + n$.

1. Find the general term formula for the sequence $\{{a_n}\}$.
2. Let ${b_n} = \frac{4a_n^2}{4a_n^2 - 1}$. Find the sum of the first $n$ terms of the sequence $\{{b_n}\}$, denoted as ${T_n}$.
3. If the inequality ${T_n} + {(-1)^{n+1}} \cdot a - n > 0$ holds for any positive integer $n$, find the range of values for the real number $a$.

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