
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
In a cube $ABCDEFGH$ with side length $s$, vertices $A$ and $G$ are diagonally opposite in the cube. $K$ and $L$ are midpoints of segments $\overline{AB}$ and $\overline{GH}$, respectively. Let $S$ be the ratio of the area of the cross-section $AKLG$ to the area of one of the faces of the cube. Calculate the value of $S^2$.

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