
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 of positive terms ${a_n}$ with the sum of the first $n$ terms denoted as $S_n$, it is known that $4S_n = a_n^2 + 4n - 1$ and $a_1 = 1$.

1. Find the general formula for the $n$-th term of the sequence ${a_n}$.
2. If ${a_n}$ is an increasing sequence and $b_n = \frac{1}{a_n \cdot a_{n+1}}$, with $T_n$ representing the sum of the first $n$ terms of the sequence ${b_n}$, find the range of real values for $m$ such that $T_n \leq \frac{m}{6}$ always holds.

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