
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]


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Consider the quadratic function $y=kx^2$ where $(x>0)$. The x-coordinate of the intersection point between the tangent line at point $(a_n,a_n^2)$ on the graph and the x-axis is denoted as $a_{n+1}$. Here, $n$ is a positive integer and $a_1=\frac{1}{3}$. Determine the value of $S_5 = \ldots$.

A) $\frac{3}{2}[1-(\frac{1}{3})^5]$

B) $\frac{1}{3}[1-(\frac{1}{3})^5]$

C) $\frac{2}{3}[1-(\frac{1}{2})^5]$

D) $\frac{3}{2}[1-(\frac{1}{2})^5]$

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