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I observed some capacitors are not in good condition on motherboard, can it be relevant to this problem?
Yes, it can be relevant. Tell me is top side of the capacitor (aluminum body with a cut line) is punctured or you can see a bubble developing on the top?
This usually happens when the capacitor ESR (Equivalent Series Resistance, every capacitor has a definite resistance as well) is getting low and the capacitor is drawing much current, itself acting as a load.
Second reason for this bubble is that capacitor is subjected to a higher voltage than its rating (which is less likely in your case, becasue were this the situation, your motherboard ICs must have burnt by now)
Electrolytic capacitors have a definite life of few thousand hours only. How old is your motherboard ?