
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 $\left\{ a_n \right\}$ satisfies $a_1+4a_2+4^2a_3+\cdots +4^{n-1}a_n=\frac{n}{4}$ $(n\in \mathbb{N}^*)$.  
(1) Find the general formula for the sequence $\left\{ a_n \right\}$.  
(2) Let $b_n=\frac{4^n a_n}{2n+1}$, find the sum of the first $n$ terms of the sequence $\left\{ b_n b_{n+1} \right\}$, denoted as $T_n$.

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