
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  $O$ ,  $A$ ,  $B$ , and  $C$  be points in space such that  $\angle AOB=60^{\circ}$ ,  $\angle BOC=90^{\circ}$ , and  $\angle COA=120^{\circ}$ . Let  $\theta$  be the acute angle between planes  $AOB$  and  $AOC$ . Given that  $\cos^2\theta=\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , compute  $100m+n$ .

*Proposed by Michael Ren*

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