
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 diagram shows a square \(PQRS\). Points \(T, U, V\), and \(W\) lie on the edges of the square, such that \(PT = 1\), \(QU = 2\), \(RV = 3\), and \(SW = 4\). The area of \(TUVW\) is half that of \(PQRS\). What is the length of \(PQ\)?

A) 5  
B) 6  
C) 7  
D) 8  
E) 9

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