
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
In the parallelogram \(ABCD\), the angle between diagonals \(AC\) and \(BD\) is \(30^\circ\). The ratio of \(AC\) to \(BD\) is given as \(2 : \sqrt{3}\). Point \(B_1\) is the reflection of vertex \(B\) with respect to line \(AC\), and point \(C_1\) is the reflection of vertex \(C\) with respect to line \(BD\). Find the ratio of the areas of triangle \(AB_1C_1\) and parallelogram \(ABCD\).

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