
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
Given that for all $x\_1 \in (0, 2]$, there exists an $x\_2 \in [1, 2]$ such that the inequality $4x\_1 \ln x\_1 - x\_1^2 + 3 + 4x\_1 x\_2^2 + 8a x\_1 x\_2 - 16x\_1 \geq 0$ holds true, find the range of values for the real number $a$.

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