
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 a quadratic function $f(x)=ax^{2}+bx$ ($a,b$ are constants, and $a\neq 0$) satisfies the condition: $f(x-1)=f(3-x)$, and the equation $f(x)=2x$ has two equal roots.  
$(1)$ Find the expression for $f(x)$.  
$(2)$ Find the maximum value of $f(x)$ on the interval $[0,t]$.

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