
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 ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, where $a_1$ = $\frac{1}{2}$, and $2S_n = S_{n-1} + 1$ ($n \geq 2, n \in \mathbb{N}^*$).

1. Find the general formula for the sequence ${a_n}$.
2. Let $b_n = \log_{\frac{1}{2}}a_n$ ($n \in \mathbb{N}^*$), and find the sum of the first $n$ terms, denoted as $T_n$, for the sequence ${\frac{1}{b_nb_{n+1}}}$.

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