
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
A positive rational number is written on a board. For any two numbers \(a\) and \(b\) already written on the board (even if they are the same), it is allowed to write the numbers \(a + 2b\), \(ab^2\), and \(a / b^2\). Will it always be possible (possibly in several steps):

a) to write the number 1 if an odd natural number is initially written? 

b) to write the number 1 if an even natural number is initially written? 

c) to write the number 1 if an arbitrary number is initially written?

d) to write the number 2 if an arbitrary number is initially written?

e) Is there a number \(x\) such that if \(x\) is written on the board, it will be possible to write any other positive rational number?

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