
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
Let $ABC$ be an acute-angled triangle. Let $D$ be the foot of the altitude corresponding to side $BC$; $M$ be the midpoint of side $BC$; and $F$ be the point where the internal angle bisector of $\angle BAC$ intersects side $BC$. Determine all triangles for which $F$ is the midpoint of segment $DM$.

Note. Determining a triangle is understood as giving the lengths of its sides, in terms of some parameter if necessary.

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