
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
For which values of the parameter \(a\) does the equation

$$
2 \log _{16}\left(2 x^{2}-x-2 a-4 a^{2}\right)-\log _{4}\left(x^{2}-a x-2 a^{2}\right)=0
$$

have two distinct roots whose sum of squares lies in the interval \((0, 4)\)?

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