
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 $f(x) = 2^x$, where $x \in \mathbb{R}$.

(Ⅰ) Solve the equation: $f(2x) - f(x+1) = 8$;

(Ⅱ) Let $a \in \mathbb{R}$, find the expression for the maximum value $M(a)$ of the function $g(x) = f(x) + a \cdot 4^x$ in the interval $[0, 1]$;

(Ⅲ) If $f(x_1) + f(x_2) = f(x_1)f(x_2)$ and $f(x_1) + f(x_2) + f(x_3) = f(x_1)f(x_2)f(x_3)$, find the maximum value of $x_3$.

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