
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
If \(q_1(x)\) and \(r_1\) are the quotient and remainder, respectively, when the polynomial \(x^6\) is divided by \(x + \frac{1}{2}\), and if \(q_2(x)\) and \(r_2\) are the quotient and remainder, respectively, when \(q_1(x)\) is divided by \(x - 1\), then what is the value of \(r_2\)?
A) $\frac{1}{32}$
B) $\frac{21}{64}$
C) $-\frac{1}{16}$
D) $\frac{21}{32}$
E) $-\frac{1}{32}$

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