
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 $f(x) = \log_{2}(x+1)$, translate its graph 1 unit to the left, then stretch all the points on the graph vertically to twice their original length (the horizontal coordinates remain unchanged) to obtain the graph of the function $y = g(x)$.

(1) Find the analytical expression and domain of the function $y = g(x)$;

(2) Find the maximum value of the function $F(x) = f(x-1) - g(x)$.

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