
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 simplifications described in exercise 547, what are the intervals between successive new moon, first quarter, full moon, last quarter, and the next new moon? (Quarter phase occurs when the centers of the Sun, Earth, and Moon form a right triangle with the vertex of the right angle at the Moon's center, resulting in a visible diameter as the shadow edge on the Moon's surface.) Assume the Earth's and the Moon's orbital radii are $150 \cdot 10^{6}$ km and $384 \cdot 10^{3}$ km respectively.

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