
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 inscribed quadrilateral \(ABCD\) bisects one of the angles from which it is drawn and also divides the opposite angle in the ratio \(1:5\). What can be the measure of angle \(\angle A\) of this quadrilateral? Express your answer in degrees. Indicate all possible options. If necessary, round the answer to hundredths or write it as a common fraction.

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