
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
Triangle  $ABC$  has  $\angle BAC=90^\circ$ . A semicircle with diameter  $XY$  is inscribed inside  $\triangle ABC$  such that it is tangent to a point  $D$  on side  $BC$ , with  $X$  on  $AB$  and  $Y$  on  $AC$ . Let  $O$  be the midpoint of  $XY$ . Given that  $AB=3$ ,  $AC=4$ , and  $AX=\tfrac{9}{4}$ , compute the length of  $AO$ .

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