
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 the inscribed quadrilateral \(ABCD\) is the angle bisector of one of the angles from which it is drawn and also divides the other angle in the ratio \(1:2\). What can be the measure of the angle \(\angle A\) of this quadrilateral? Express the answer in degrees. List all possible values. If necessary, round the answer to the hundredths or express it as a common fraction.

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