
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
Since \( \frac{1}{3} < \frac{3}{8} \), the possible values for \( \xi \) are \( \xi = 4 \) or \( \xi = 9 \).

Case 1: \( \xi = 4 \)

Transforming the equation:
\[ 4a x^{2} - 16x + a4 = 0 \rightarrow a x^{2} - 4x + a = 0 \rightarrow a(x^{2} + 1) = 4x \rightarrow a = \frac{4x}{x^{2} + 1} \]

The derivative of the function \( a = \frac{4x}{x^{2} + 1} \) is \( a' = \frac{4(1 - x^{2})}{(x^{2} + 1)^{2}} \leq 0 \) on the interval \([1, 2]\), so the set of values for \( a \) for which the equation has at least one solution on \([1, 2]\) is \([a(2), a(1)] = [\frac{8}{5}, 2]\).

Case 2: \( \xi = 9 \)

Transforming the equation:
\[ 4a x^{2} - 36x + 9a = 0 \rightarrow a = \frac{36x}{4x^{2} + 9} \]

The derivative of the function \( a = \frac{36x}{4x^{2} + 9} \), which is \( a' = \frac{36(9 - 4x^{2})}{(4x^{2} + 9)^{2}} \), is positive on the interval \([1, \frac{3}{2}]\) and negative on the interval \([\frac{3}{2}, 2]\). The value range of the function \( a = \frac{36x}{4x^{2} + 9} \) on the interval \([1, 2]\) is \([a(1), a(\frac{3}{2})] = [\frac{36}{13}, 3]\).

Combining cases 1 and 2, we get \( a \in [\frac{8}{5}, 2] \cup [\frac{36}{13}, 3] \).

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