
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 the sequence $1$, $1$, $2$, $1$, $2$, $4$, $1$, $2$, $4$, $8$, $1$, $2$, $4$, $8$, $16$, $\ldots$, where the first term is $2^{0}$, the next two terms are $2^{0}$, $2^{1}$, and the following three terms are $2^{0}$, $2^{1}$, $2^{2}$, and so on.<br/>$(1)$ The $100$th term of this sequence is ______；<br/>$(2)$ An integer $N$ satisfies the condition: $N \gt 1000$ and the sum of the first $N$ terms of this sequence is a power of $2$, then the smallest integer $N=\_\_\_\_\_\_$.

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