
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
The mathematical giant Euler was the first to represent a polynomial in terms of $x$ using the notation $f(x)$. For example, when $x=-1$, the value of the polynomial $f(x)=x^{2}+2x-3$ is denoted as $f(-1)$, then $f(-1)=(-1)^{2}+2\times (-1)-3=1+(-2)-3=-4$. Given $f(x)=ax^{5}+bx^{3}+4x+c$. Please solve the following problems:<br/>$(1)$ If $f(0)=6$, then $c=$______；<br/>$(2)$ If $f(0)=-2$, and $f(1)=5$, find the value of $f(-1)$；<br/>$(3)$ If $f(5)+f(-5)=6$, and $f(2)=8$, find the value of $f(-2)$.

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