
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 $f(x) = a^{x} + b^{x}$ $(a > 0, b > 0, a \neq 1, b \neq 1)$:
1. Assume $a=2$, $b=\frac{1}{2}$:
   a. Find the root of the equation $f(x) = 2$.
   b. If for any $x \in \mathbb{R}$, the inequality $f(2x) \geq mf(x) - 6$ always holds, find the maximum value of the real number $m$.
2. If $0 < a < 1$, $b > 1$, and the function $g(x) = f(x) - 2$ has only one zero, find the value of $ab$.

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