
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

Given points \( A(1,1) \), \( B\left(\frac{1}{2},0\right) \), and \( C\left(\frac{3}{2},0\right) \), the plane region \( G \) is enclosed by the lines passing through points \( A \) and \( B \), through points \( A \) and \( C \), and the line \( y=a \) where \( 0 < a < 1 \). If the probability of any point in the rectangular region \(\{(x, y) \mid 0<x<2, 0<y<1\}\) falling into the region \( G \) is \(\frac{1}{16}\), find the value of \( a \).

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