
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

Two triangles \(A_{1} B_{1} C_{1}\) and \(A_{2} B_{2} C_{2}\), whose areas are \(S_{1}\) and \(S_{2}\) respectively, are positioned so that the rays \(A_{1} B_{1}\) and \(A_{2} B_{2}\), \(B_{1} C_{1}\) and \(B_{2} C_{2}\), \(C_{1} A_{1}\) and \(C_{2} A_{2}\) are oppositely directed. Find the area of the triangle with vertices at the midpoints of the segments \(A_{1} A_{2}, B_{1} B_{2}, C_{1} C_{2}\).

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