
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 quadratic function $f(x)=ax^{2}+bx$ that satisfies: $(1)$ $f(2)=0$, $(2)$ the equation $f(x)=x$ has two equal real roots with respect to $x$.
$(1)$ Find the analytic expression of the function $f(x)$;
$(2)$ Find the range of the function $f(x)$ on $[0,3]$.

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