
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 the line $l\_n$: $y = x - \sqrt{2n}$ and the circle $C\_n$: $x^2 + y^2 = 2a\_n + n$, where $n \in \mathbb{N}^+$. They intersect at two distinct points $A\_n$ and $B\_n$. The sequence ${a\_n}$ satisfies: $a\_1 = 1$, $a\_{n+1} = \frac{1}{4}|A\_nB\_n|^2$.

1. Find the general term formula $a\_n$ for the sequence ${a\_n}$.
2. If $b\_n = \frac{n}{4a\_n}$, find the sum of the first $n$ terms, $T\_n$, for the sequence ${b\_n}$.

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