
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 $\{a_n\}$ where $a_1 = 1$ and $a_{n+1} = \frac{a_n}{a_n + 4}$ for $n \in \mathbb{N}^*$.
(1) Find the values of $a_2$ and $a_3$;
(2) Find the general term formula $a_n$ for the sequence $\{a_n\}$;
(3) Let $b_n = (4^n - 1)\cdot \frac{n}{2^n}\cdot a_n$, and denote its summation of the first $n$ terms as $T_n$. If the inequality $2^{n-1} \lambda < 2^{n-1}T_n + \frac{3n}{2}$ holds for all $n \in \mathbb{N}^*$, determine the range of values for $\lambda$.

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