
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 defined by \(a_1=1\) and \(a_k=\left\lfloor\sqrt{a_1 + a_2 + \ldots + a_{k-1}}\right\rfloor\), find \(a_{1000}\). Note: \(\left\lfloor \cdot \right\rfloor\) denotes the floor function, which returns the greatest integer less than or equal to the expression inside.

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