
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 a quadratic function $f(x) = ax^2 + bx$ that satisfies $f(x-1) = f(x) + x - 1$.
(Ⅰ) Find the explicit expression of $f(x)$.
(Ⅱ) Find the zeros of the function $f(x)$ and write the set of values for $x$ when $f(x) < 0$.
(Ⅲ) Let $F(x) = 4f(a^x) + 3a^{2x} - 1$ (with $a>0$ and $a \neq 1$). If $F(x)$ has a maximum value of 14 for $x \in [-1, 1]$, find the value of $a$.

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