
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
The edge length of cube \( A B C D A_{1} B_{1} C_{1} D_{1} \) is \( a \). Points \( E \) and \( F \) are the midpoints of edges \( B B_{1} \) and \( C C_{1} \) respectively. Consider triangles with vertices at the intersection points of a plane parallel to plane \( A B C D \) with lines \( A C_{1}, C E, \) and \( D F \). Find the minimum area of these triangles.

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