n lamps problem

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We have 14a5fd3da3f4c1da4f9747b18896a66b7e5ac54c lamps 8d4aa95a18d34721de92a9a458d5c417a6ab01e3 in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp 03b0356eb36de0491323c214843a4e3e73bd81bd and its neighbours (only one neighbour for cfca8d4a5ff8f41fb9090681b1bf78002b651ad7 or cf07c169c82b9b343ffa612cc225394b47d6732a, two neighbours for other a95cf1baf338669747fa0f7c114f3d29f70ba7d8) are in the same state, then 03b0356eb36de0491323c214843a4e3e73bd81bd is switched off; – otherwise, 03b0356eb36de0491323c214843a4e3e73bd81bd is switched on.
Initially all the lamps are off except the leftmost one which is on.

89cdb71aab82773f558609204bacae3278fb1343 Prove that there are infinitely many integers 6d3f8b726378d5420223c5cb14b10f24b202b187 for which all the lamps will eventually be off.
25358830cda03ffcc7b0edd6903f86be71cc9dab Prove that there are infinitely many integers 6d3f8b726378d5420223c5cb14b10f24b202b187 for which the lamps will never be all off.

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