
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
According to Archimedes' calculations, the ratio of the circumference of a circle to its diameter - which we denote as $\pi$ based on Euler's recommendation - is close to both $3 + \frac{10}{71} = \frac{223}{71}$ and $3 + \frac{1}{7} = \frac{22}{7}$. The former is greater and the latter is smaller. This implies that if we measure the chord of a circle with a diameter of 7 units 22 times consecutively, the endpoint of the last chord will be close to the starting point of the first chord. Calculate the central angle corresponding to the arc between the starting and ending points. Similarly, carry out the calculations starting from the approximation $\frac{223}{71}$. Based on these calculations, can we determine which of the two approximations is better? (Only a four-digit logarithm table can be used; it is advisable to calculate using the $\lg \sin$ table because the difference between adjacent data in that table is larger.)

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