
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  $T$  be the triangle in the  $xy$ -plane with vertices  $(0, 0)$ ,  $(3, 0)$ , and  $\left(0, \frac32\right)$ . Let  $E$  be the ellipse inscribed in  $T$  which meets each side of  $T$  at its midpoint. Find the distance from the center of  $E$  to  $(0, 0)$ .

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