Extrapolation of sinusoidal signal
One natural way to do this is to use the the total $2N$ samples (two chunks of $N$ samples each) to estimate the parameters. Amplitude, frequency, and phase estimation are all textbook equations that you can look up so the problem here is the fact that they are all mixed together. If you are assuming that you don't know how many sinusoids are mixed, then you also have to detect how many are present. The basic pseudo code is:
Detect how many sinusoids there are
For each sinusoid:
    Estimate amplitude
    Estimate frequency
    Estimate phase
end
Reconstruct signal at n = N + 10000

Detect how many sinusoids there are
You can do this by detecting how many peaks there are in the FFT of the signal. A sinusoid will present a peak at its frequency. At a peak, the slope of the curve will change from positive to negative, so the slope has a zero crossing at the peak. For a discrete signal, there are different ways to compute the slope/derivative but I would start with the simple before moving to more sophisticated methods. A simple way to compute the derivative is just the difference, $x_d[n]=x[n]-x[n-1]$, and a zero crossing happens if $x_d[n-1] &gt; 0 &gt; x_d[n]$. This will give you the number of zero crossings and where they occur.
Edit
The model that was presented in the question is a noise free model, and the detection described relies on that. You can still use this zero crossing method for a noisy signal but need to apply a threshold because the noise will cause many small peaks to occur. The threshold is needed to keep only the ones that have a significant enough amount of energy. Threshold should be picked to satisfy some probability of false alarm metric that you desire.
Estimate ...
As I mentioned above, estimating the amplitude, frequency, and phase of a sinusoid can be found through the FFT peak finding method (and we already found the locations of the peaks) so we are all set.
Reconstruct signal
Now that we have all the estimates, you can calculate your equation using your estimates.
$$ \hat{s}(n+10000) = \sum_i \hat{A}_i \text{sin}\bigg( \frac{2\pi \hat{f}_i (n+10000)}{f_s} + \hat{\theta}_i \bigg) $$