
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 sidereal month of the Moon (time for the Moon to return to the same position with respect to the stars) is 27 days 7 hours 43 minutes.
- The sidereal year of the Earth (time for the Earth to complete one orbit around the Sun) is 365 days 6 hours 9 minutes.

Calculate the synodic month (the period between successive similar phases of the Moon, such as full moon to full moon) assuming that:

- The Earth's orbit around the Sun and the Moon’s orbit around the Earth are circular.
- The movements are uniform.
- The planes of the two orbits coincide.

Also, determine whether the same interval applies to new moons, first quarters, and last quarters.

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