
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 ancient China, when determining the pentatonic scale of Gong, Shang, Jiao, Zhi, and Yu, a method of calculation involving a three-part loss and gain was initially used. The second note was obtained by subtracting one-third from the first note, and the third note was obtained by adding one-third to the second note, and so on until all five notes were obtained. For example, if the length of an instrument capable of producing the first fundamental note is 81, then the length of an instrument capable of producing the second fundamental note would be $81 \times (1-\frac{1}{3}) = 54$, and the length of an instrument capable of producing the third fundamental note would be $54 \times (1+\frac{1}{3}) = 72$, that is, decreasing by one-third first and then increasing by one-third. If the length of an instrument capable of producing the first fundamental note is $a$, and the length of an instrument capable of producing the fourth fundamental note is 32, then the value of $a$ is ____.

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