
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
Let the function
$$
f(x)=\log _{2}\left(a^{x}-b^{x}\right),
$$
and it is given that $f(1)=1$ and $f(2)=\log _{2} 12$.
(1) Find the values of $a$ and $b$.
(2) When $x \in[1,2]$, find the maximum value of $f(x)$.

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