
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
Each diagonal of a cyclic quadrilateral $ABCD$ bisects one of the angles from which it is drawn and also divides the other angle in a ratio of $1:3$. What is the possible measure of angle $\angle A$ of this quadrilateral? Express the answer in degrees. List all possible values. If necessary, round the answer to two decimal places or write it as a common fraction.

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