
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 function $f(x) = a^x$ ($a > 0$, $a \neq 1$) has its maximum value on the interval $[-1, 2]$ being 8 times its minimum value.
(Ⅰ) Find the value of $a$;
(Ⅱ) When $a > 1$, solve the inequality $\log_a(2a + 2x) < \log_a(x^2 + 1)$.

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