
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 as $S_n$, given that $a_1=2$, $a_2=8$, and $S_{n+1}+4S_{n-1}=5S_n$ (for $n\geq2$). 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 $n$ such that $(1- \frac{1}{T_2})(1- \frac{1}{T_3})\ldots(1- \frac{1}{T_n}) > \frac{51}{101}$.

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