
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
Consider the polynomial $(ax + by)^7$ expanded in decreasing powers of $x$. The third and fourth terms are found to have equal values when evaluated at $x=a$ and $y=b$, where $a$ and $b$ are positive numbers whose sum equals one. Determine the value of $a$.
A) $\frac{\sqrt{15}}{5}$  
B) $\frac{5}{\sqrt{15}}$  
C) $\frac{4}{5 + \sqrt{15}}$  
D) $\frac{5}{5 + \sqrt{15}}$  
E) $\frac{\sqrt{15}}{15}$

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