
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
Let  $f(x)=x^3+ax^2+bx+c$  and  $g(x)=x^3+bx^2+cx+a$ , where  $a,b,c$  are integers with  $c\not=0$ . Suppose that the following conditions hold:

[list=a][*] $f(1)=0$ ,
[*]the roots of  $g(x)=0$  are the squares of the roots of  $f(x)=0$ .[/list]

Find the value of  $a^{2013}+b^{2013}+c^{2013}$ .

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