
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 $A_{2} B_{2} C_{2}$, are positioned such that the sides $A_{1} B_{1}$ and $A_{2} B_{2}$, $B_{1} C_{1}$ and $B_{2} C_{2}$, and $A_{1} C_{1}$ and $A_{2} C_{2}$ are parallel respectively. Connect vertex $A_{1}$ with vertices $B_{2}$ and $C_{2}$, then connect vertex $B_{1}$ with vertices $C_{2}$ and $A_{2}$, and finally connect vertex $C_{1}$ with vertices $A_{2}$ and $B_{2}$. What can be the area of the hexagon determined by the midpoints of these connecting segments, if the areas of the original triangles are $T_{1}$ and $T_{2}$?

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