
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
Consider a sequence $\{a_n\}$ with the following pattern: $2^1$, $2^2$, $2^2$, $2^3$, $2^3$, $2^3$, $2^4$, $2^4$, $2^4$, $2^4$, and so on, where $k$ instances of $2^k$ are repeated. The rule is that when $\frac{k(k-1)}{2} < n \leq \frac{k(k+1)}{2}$ (where $k$ is a positive integer), $a_n = 2^k$. Let $S_n$ denote the sum of the first $n$ terms of $\{a_n\}$.
(Ⅰ) Find $a_{50}$.
(Ⅱ) Find $S_{50}$.

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