
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 quadratic function $f(x) = mx^2 - 2x - 3$, if the solution set of the inequality $f(x) < 0$ is $(-1, n)$.
(1) Solve the inequality about $x$: $2x^2 - 4x + n > (m + 1)x - 1$;
(2) Determine whether there exists a real number $a \in (0, 1)$, such that the minimum value of the function $y = f(a^x) - 4a^{x+1}$ ($x \in [1, 2]$) is $-4$. If it exists, find the value of $a$; if not, explain why.

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