
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
On the sides $AB$, $BC$, and $CA$ of an equilateral triangle $ABC$, find the points $X$, $Y$, and $Z$ (respectively) such that the area of the triangle formed by the lines $CX$, $BZ$, and $AY$ is one-fourth the area of triangle $ABC$, and so that the following condition holds: \(\frac{\mathrm{AX}}{\mathrm{XB}} = \frac{\mathrm{BY}}{\mathrm{YC}} = \frac{\mathrm{CZ}}{\mathrm{ZA}}.\)

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