
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 midpoints of the sides \(PQ\), \(QR\), and \(RP\) of triangle \(PQR\) are \(A\), \(B\), and \(C\) respectively. The triangle \(ABC\) is enlarged from its centroid \(S\) by a factor of \(k\), where \(1 < k < 4\). The sides of the resulting enlarged triangle intersect the sides of triangle \(PQR\) at points \(D_{1}D_{2}\), \(E_{1}E_{2}\), and \(F_{1}F_{2}\) respectively. For which value of \(k\) will the area of the hexagon \(D_{1}D_{2}E_{1}E_{2}F_{1}F_{2}\) be twice the area of triangle \(ABC\)?

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