
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 that the sum of the first $n$ terms of a sequence $\{a_{n}\}$ is $S_{n}$, $a_{1}=4$, and ${S}_{n+1}=2{S}_{n}+{a}_{1}$ for $n\in\mathbb{N}^{*}$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ Let the product of the first $n$ terms of the sequence $\{a_{n}\}$ be $T_{n}$. Find the maximum value of $n$ when $\log _{2}T_{n}\leqslant 20$.

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