
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
In the ancient Chinese mathematical work "Jiuzhang Suanshu," there is a problem as follows: "There are plants named Pu (a type of aquatic plant) and Guan (commonly known as water onion or mat grass). Pu grows 3 feet in one day, and Guan grows 1 foot in one day. From the next day on, the growth rate of Pu is halved daily, while the growth rate of Guan doubles daily. How many days will it take for their lengths to be equal?" The essence of the problem is: Initially, Pu grows 3 feet in one day, and Guan grows 1 foot in one day. From the next day on, the growth rate of Pu is halved daily, and the growth rate of Guan doubles daily. If the lengths of Pu and Guan are to be equal, the required time is approximately ____ days. (The result should be kept to one decimal place, reference data: $\lg 2 \approx 0.30$, $\lg 3 \approx 0.48$)

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