
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 \( A_1,A_2,\dots \) lie on the \( x \)-axis, and distinct points \( B_1,B_2,\dots \) lie on the graph of \( y=x^{\frac{2}{3}} \). For every positive integer \( n \), \( A_{n-1}B_nA_n \) is an equilateral triangle. What is the least \( n \) for which the length \( A_0A_n \geq 150 \)?

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