
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
The "Tiaorizhi" method, invented by mathematician He Chengtian during the Southern and Northern Dynasties, is a programmatic way to seek precise fractional representations of approximate values. Its theoretical basis is: if $\frac {b}{a}$ < $\frac {d}{c}$ (where a, b, c, d ∈ ℕ*), then $\frac {b}{a}$ < $\frac {b+d}{a+c}$ < $\frac {d}{c}$. For example, for π = 3.14..., $\frac {3}{1}$ < π < $\frac {7}{2}$, and using the "Tiaorizhi" method once gives the fraction $\frac {10}{3}$, narrowing the range to $\frac {3}{1}$ < π < $\frac {10}{3}$. If we want the error between the approximate value and π to be less than 0.1, we need to use the "Tiaorizhi" method at least \_\_\_\_\_\_ times, and the corresponding approximate fraction for π is \_\_\_\_\_\_.

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