
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 $Q$ and $E$ of triangle $M Q E$ and intersects its sides $M Q$ and $M E$ at points $B$ and $D$ respectively, which are different from the vertices of the triangle. The ratio of the area of triangle $B D M$ to the area of triangle $M Q E$ is $\frac{9}{121}$.

a) Find the ratio $Q E : B D$.

b) Additionally, it is known that the ratio of the areas of triangles $B M E$ and $D Q M$ is 4. Find the ratio $B Q : D E$.

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