
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
Triangle $ABC$ is an isosceles triangle with $BC$ as its base. Let $F$ be the midpoint of $BC$. Point $D$ is an internal point on segment $BF$. The perpendicular to $BC$ at point $D$ intersects side $AB$ at $M$, and a line passing through point $D$ parallel to $AB$ intersects side $AC$ at $P$. Express the ratio of the areas of triangles $AMP$ and $ABC$ as a function of $k = BD : BC$.

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