
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
A closed glass container in the shape of a cube with edge length 1 has a base labeled $A B C D$, and vertical edges labeled $A A_{1}, B B_{1}, C C_{1}$, and $D D_{1}$. The container is filled with water up to $\frac{2}{3}$ of its height, with air above. The container is positioned so that the body diagonal $A C_{1}$ is vertical (with $A$ at the bottom). At which points on the edges will the water surface intersect? Provide the positions of these points as approximations rounded to two decimal places.

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