
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 sequence $\{a_n\}$ satisfies 
$$a_1 + 4a_2 + 4^2a_3 + \ldots + 4^{n-1}a_n = \dfrac{n}{4} \quad (n \in \mathbb{N^*}).$$
(1) Find the general term formula for the sequence $\{a_n\}$;  
(2) Let $b_n = \dfrac{4^n a_n}{2n+1}$, find the sum of the first $n$ terms $T_n$ of the sequence $\{b_n b_{n+1}\}$.

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