
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 $\left\{a_{n}\right\}$ where all terms are positive numbers, and the sum of the first $n$ terms is $S_{n}$, with $a_{1}=1$.
1. If the sequence $\left\{a_{n}\right\}$ is an arithmetic progression with $S_{10}=70$, find the general formula for the sequence $\left\{a_{n}\right\}$.
2. If the sequence $\left\{a_{n}\right\}$ is a geometric progression with $a_{4}=\dfrac{1}{8}$, determine the smallest value of $n$ that satisfies $S_{n} \gt 100a_{n}$.

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