
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 function \( f(x) = x^3 \), the tangent to the curve at the point \( (a_k, f(a_k)) \) (where \( k \in \mathbf{N}^{*} \)) intersects the x-axis at \( (a_{k+1}, 0) \). If \( a_1 = 1 \), find the value of \[ \frac{f\left(\sqrt[3]{a_1}\right) + f\left(\sqrt[3]{a_2}\right) + \cdots + f\left(\sqrt[3]{a_{10}}\right)}{1 - \left(\frac{2}{3}\right)^{10}}. \]

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