
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 a sequence $\{a_n\}$ with the sum of the first $n$ terms denoted by $S_n$, and $S_n = 2a_n - 2$,

$(1)$ Find the general formula for the term $a_n$;  
$(2)$ Let $b_n = a_n \log_2 a_n$ and $T_n = b_1 + b_2 + ... + b_n$, find the smallest positive integer $n$ such that $T_n - n \cdot 2^{n+1} + 50 < 0$.

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