
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
Let \( A_1A_2A_3A_4A_5A_6 \) be a convex hexagon such that \( A_iA_{i+2} \parallel A_{i+3}A_{i+5} \) for \( i=1,2,3 \) (we take \( A_{i+6}=A_i \) for each \( i \) ). Segment \( A_iA_{i+2} \) intersects segment \( A_{i+1}A_{i+3} \) at \( B_i \), for \( 1 \leq i \leq 6 \), as shown. Furthermore, suppose that \( \triangle A_1 A_3 A_5 \cong \triangle A_4 A_6 A_2 \). Given that \([A_1 B_5 B_6] = 1\), \([A_2 B_6 B_1] = 4\), and \([A_3 B_1 B_2] = 9 \) (by \([XYZ]\) we mean the area of \( \triangle XYZ \)), determine the area of hexagon \( B_1 B_2 B_3 B_4 B_5 B_6 \).

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