
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 $\{a\_n\}$, if $a\_1=0$ and $a\_i=k^2$ ($i \in \mathbb{N}^*, 2^k \leqslant i < 2^{k+1}, k=1,2,3,...$), find the smallest value of $i$ that satisfies $a\_i + a_{2i} \geq 100$.

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