
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 tetrahedron with  $AB=CD=1300$ ,  $BC=AD=1400$ , and  $CA=BD=1500$ . Let  $O$  and  $I$  be the centers of the circumscribed sphere and inscribed sphere of  $ABCD$ , respectively. Compute the smallest integer greater than the length of  $OI$ .

*Proposed by Michael Ren*

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