
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  $ABCD$  be a rectangle of sides  $AB = 4$  and  $BC = 3$ . The perpendicular on the diagonal  $BD$  drawn from   $A$  cuts  $BD$  at point  $H$ . We call  $M$  the midpoint of  $BH$  and  $N$  the midpoint of  $CD$ . Calculate the measure of the  segment  $MN$ .

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