
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 a triangle \(ABC\) and a plane \(S\), construct a line \(g\) passing through point \(A\) that is parallel to the plane \(S\). Let \(B_1\) and \(C_1\) be the projections of points \(B\) and \(C\) on this line. The requirement is that \(\overline{AB_1} = 2 \cdot \overline{AC_1}\).

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