
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 real numbers  $a_1,a_2,a_3$  and  $b{}$  are given. The equation \[(x-a_1)(x-a_2)(x-a_3)=b\]has three distinct real roots,  $c_1,c_2,c_3.$  Determine the roots of the equation \[(x+c_1)(x+c_2)(x+c_3)=b.\]*Proposed by A. Antropov and K. Sukhov*

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