
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
Given the conditions: $\angle A+\angle B=\angle C$, $\angle A:\angle B:\angle C=1:2:3$, $\angle A=90^{\circ}-\angle B$, $\angle A=\angle B=\frac{1}{2}\angle C$, $\angle A=2\angle B=3\angle C$, determine the number of conditions that can uniquely define a right triangle.

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