
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
A circle passes through the vertices \( P \) and \( T \) of triangle \( M P T \) and intersects its sides \( M P \) and \( M T \) at points \( D \) and \( E \), respectively, distinct from the vertices of the triangle. The ratio of the area of triangle \( M D E \) to the area of triangle \( M P T \) is \( \frac{1}{4} \).

a) Find the ratio \( D E: T P \).

b) Suppose it is additionally known that the ratio of the areas of triangles \( M D T \) and \( M E P \) is \( \frac{4}{9} \). Find the ratio \( T E: P D \).

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