
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
Suppose in triangle $ABC$, vertex $A$ is at $(0,0)$, $B$ is at $(b, 0)$, and $C$ is at $(c_x, c_y)$. A line segment begins at $A$ with length zero and travels in a straight line towards $C$. Additionally, when the segment passes through the centroid of the triangle $ABC$, its length is maximized. Determine the maximum length of the segment when it is at the centroid of $ABC$.
A) $\sqrt{\left(\frac{b+c_x}{2}\right)^2 + \left(\frac{c_y}{2}\right)^2}$
B) $\sqrt{\left(\frac{b+c_x}{4}\right)^2 + \left(\frac{c_y}{4}\right)^2}$
C) $\sqrt{\left(\frac{b+c_x}{3}\right)^2 + \left(\frac{c_y}{3}\right)^2}$
D) $\sqrt{b^2 + c_x^2 + c_y^2}$

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