
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 $S_n=2^{n+1}$, and $b_n=log_{2}( a_{ n }^{ 2 }\cdot 2^{a_{n}})$, the sum of the first n terms of the sequence ${b_n}$ is $T_n$. Find the smallest value of n that satisfies $T_n>1024$.

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