
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 numbers \( m \) and \( k \) satisfy the equations \( 2^{m} + 2^{k} = p \) and \( 2^{m} - 2^{k} = q \). What is the value of \( 2^{m+k} \) in terms of \( p \) and \( q \)?

A) \( \frac{p^2 - q^2}{4} \)

B) \( \frac{pq}{2} \)

C) \( p + q \)

D) \( \frac{(p - q)^2}{4} \)

E) \( \frac{p + q}{p - q} \)

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