
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
Let \(A_0=(0,0)\). Distinct points \(A_1, A_2, \ldots\) lie on the \(x\)-axis, and distinct points \(B_1, B_2, \ldots\) lie on the graph of \(y=x^2\). For every positive integer \(n\), \(A_{n-1}B_nA_n\) is an equilateral triangle. What is the least \(n\) for which the length \(A_0A_n \geq 50\)?

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