
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 $\omega$ with radius 6 and center $O$ is inscribed in an acute-angled triangle $CFM$ and touches its sides $CM$ and $FM$ at points $P$ and $K$, respectively. A circle $\Omega$ with radius $\frac{5 \sqrt{13}}{2}$ and center $T$ is circumscribed about triangle $PKM$.

a) Find $OM$.

b) Additionally, suppose the ratio of the area of triangle $CFT$ to the area of triangle $CFM$ is $\frac{5}{8}$. Find the length of the angle bisector $MA$ of triangle $CFM$, as well as its area.

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