
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
The vertices of  $\triangle ABC$  are labeled in counter-clockwise order, and its sides have lengths  $CA = 2022$ ,  $AB = 2023$ , and  $BC = 2024$ .  Rotate  $B$   $90^\circ$  counter-clockwise about  $A$  to get a point  $B'$ .  Let  $D$  be the orthogonal projection of  $B'$  unto line  $AC$ , and let  $M$  be the midpoint of line segment  $BB'$ .  Then ray  $BM$  intersects the circumcircle of  $\triangle CDM$  at a point  $N \neq M$ .  Compute  $MN$ .

*Proposed by Thomas Lam*

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