
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
(1) The graph of the quadratic function $h(x) = x^2$ is translated 1 unit to the right and 2 units downward to obtain the graph of the function $f(x)$. Write down the explicit formula for $f(x)$, and determine the range of $f(x)$ when $x \in [0,4]$.
(2) Find the minimum value of $f(x) = x^2 - 2ax - 1$ on the interval $[0,2]$.

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