
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 $f(x)=(x^{2}-6x+c_{1})(x^{2}-6x+c_{2})(x^{2}-6x+c_{3})$, where the set of solutions to $f(x)=0$ is $M={x|f(x)=0}={x_{1},x_{2},x_{3},x_{4},x_{5}}⊆N^{*}$, and $c_{1}≥c_{2}≥c_{3}$, determine the value of $c_{1}-c_{3}$.

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