
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 a rectangle \(ABCD\). A line through point \(A\) intersects segment \(CD\) at point \(X\) such that the areas of the resulting shapes satisfy \(S_{AXD}: S_{ABCX} = 1:2\). A line through point \(X\) intersects segment \(AB\) at point \(Y\) such that \(S_{AXY}: S_{YBCX} = 1:2\). Finally, a line through point \(Y\) intersects segment \(XC\) at point \(Z\) such that \(S_{XYZ}: S_{YBCZ} = 1:2\).

Calculate the ratio of the areas \(S_{AXD}: S_{AXZY}\).

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