
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 $L$ and $M$ of triangle $FLM$ and intersects its sides $FL$ and $FM$ at points $A$ and $H$ respectively, different from the vertices of the triangle. The ratio of the area of triangle $FLM$ to the area of triangle $AFH$ is $\frac{49}{9}$.

a) Find the ratio $LM : AH$.

b) Additionally, it is known that the ratio of the areas of triangles $AFM$ and $FHL$ is $\frac{1}{4}$. Find the ratio $AL : MH$.

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