
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 $S_{n}$ be the sum of the first $n$ terms of the sequence ${a_{n}}$, with $a_{1}=2$, $a_{2}=8$, and $S_{n+1}+4S_{n-1}=5S_{n}$ for $n\geq 2$. Additionally, let $T_{n}$ be the sum of the first $n$ terms of the sequence ${\log_{2}a_{n}}$.

(1) Find the general term formula for the sequence ${a_{n}}$.
(2) Find the largest positive integer value of $n$ such that $(1-\frac{1}{T_{2}})(1-\frac{1}{T_{3}})...(1-\frac{1}{T_{n}})\geq \frac{1009}{2016}$.

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