
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
During the solar eclipse on May 20, 1966, the so-called maximum phase was 0.766 for observers in Battonya, and 0.678 for observers in Sopron. (This means that at the moment of maximum darkness, the given proportion of the Sun's diameter, which falls on the common symmetry axis of the Sun and Moon discs, is not visible.) Calculate for these situations the ratio of the covered area to the total area of the Sun's disc (understood on an appropriate photographic record). The angular diameters of the Sun and the Moon on this day are $31^{\prime} 39^{\prime \prime}$ and $31^{\prime} 6^{\prime \prime}$, respectively.

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