
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
Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$. Given that $a_1=2$, $a_2=8$, and $S_{n+1}+4S_{n-1}=5S_n$ ($n\geqslant 2$), let $T_n$ be the sum of the first $n$ terms of the sequence $\{\log_2 a_n\}$.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) Find the maximum positive integer value of $n$ that satisfies $\left( 1- \frac{1}{T_2} \right)\left( 1- \frac{1}{T_3} \right)\cdots\left( 1- \frac{1}{T_n} \right)\geqslant \frac{1011}{2018}$.

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