
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 geometric sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and it is known that $6S_n = 3^{n+1} + a$ ($n \in \mathbb{N}^*$).

(1) Find the value of $a$ and the general formula for the sequence $\{a_n\}$;

(2) If $b_n = (1 - an) \log_{3}(a_n^2 \cdot a_{n+1})$, find the sum of the first $n$ terms, $T_n$, of the sequence $\left\{ \frac{1}{b_n} \right\}$.

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