
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 \( M \) and \( N \) be two points on the Thales' circle of segment \( AB \), distinct from \( A \) and \( B \). Let \( C \) be the midpoint of segment \( NA \), and \( D \) be the midpoint of segment \( NB \). The circle is intersected at the point \( E \) a second time by the line \( MC \), and at point \( F \) by the line \( MD \). What is the value of the expression

\[ 
MC \cdot CE + MD \cdot DF 
\]

if \( AB = 2 \) units?

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