
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
Through a point $P$ inside the triangle $\triangle ABC$, a line is drawn parallel to the base $AB$, dividing the triangle into two regions where the area of the smaller region (above the line) to the larger region (below the line) is in the ratio 1:3. If the altitude to $AB$ has a length of $1$, find the distance from $P$ to $AB$.

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