
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 the functions f(x) = x and g(x) = ax^2 - x, where a > 0. If for all x1 in the interval [1, 2], there exists an x2 in the interval [1, 2] such that f(x1) * f(x2) = g(x1) * g(x2) holds true, then find the value of a.

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