
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
Suppose \( x \) and \( y \) are inversely proportional, and \( z \) and \( w \) are directly proportional. Initially, \( x = 2 \) and \( y = 5 \), such that \( xy = 10 \), and \( z = 3 \) with \( w = 9 \), giving \( z \propto w \). If \( x \) is increased by 20%, find the new values of \( y \) and \( w \) when \( z \) is doubled.
**A)** \( y' = 4.167 \) and \( w' = 14 \)
**B)** \( y' = 4.2 \) and \( w' = 18 \)
**C)** \( y' = 4.167 \) and \( w' = 18 \)
**D)** \( y' = 4.5 \) and \( w' = 17 \)

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