
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 $f(x)=-x^{3}+ax^{2}+bx+c$ and the tangent line at point $P(1,f(1))$ on the graph of $f(x)$ is $y=-3x+1$, and the function $g(x)=f(x)-ax^{2}+3$ is an odd function.

$(1)$ Find the expression for the function $f(x)$;

$(2)$ Find the extreme values of the function $f(x)$.

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