I struggle to write and solve Lagrangian in the Ramsey model.

I have the following individual preferences with a budget constraint:

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and an additional constraint that is a no-ponzi condition:

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How to state a Lagrangian in this problem and solve FOC wrt. c_t, a_t+1 and a_T+1???

I tried to combine the two budget constraints:

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and then my Langrangian looks like this:

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But, I´m not sure that´s correct, because my task says that additional Lagrangian multiplier should be added......


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