
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 an isosceles triangle $ABC$ with $AB=AC$, a line through a point $P$ inside the triangle, parallel to the base $BC$, divides the triangle into two regions with areas in the ratio of $1:3$. If the altitude from $A$ to $BC$ is $4$ units, find the distance from $P$ to $BC$.

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