equation solving - Parsing output of a reduce operation

This question is a broader question than this. The output of Reduce can be of different forms. The solutions proposed for that question works well when the output of Reduce is as provided in that question. The output of Reduce can also be of the forms:

$n_2\in \mathbb{Z}\land n_1=1\land 0\leq n_2\leq 1993.$

or

$\left(n_1|n_2\right)\in \mathbb{Z}\land \left(\left(n_1=0\land 1.\leq n_2\leq 19979.\right)\lor \left(1.\leq n_1\leq 2222.\land 0\leq n_2\leq \text{3.5534074004528205$\grave{ }$*${}^{\wedge}$-79} \sqrt{-4.55384\times 10^{157} n_1^2+1.50383\times 10^{162} n_1+7.9031\times 10^{164}}+\text{1.7152185859604652$\grave{ }$*${}^{\wedge}$-56} \left(3.20659\times 10^{56} n_1+5.82404\times 10^{59}\right)\right)\lor \left(2223.\leq n_1\leq 33540.\land \text{1.7152185859604652$\grave{ }$*${}^{\wedge}$-56} \left(3.20659\times 10^{56} n_1+5.82404\times 10^{59}\right)-\text{3.5534074004528205$\grave{ }$*${}^{\wedge}$-79} \sqrt{-4.55384\times 10^{157} n_1^2+1.50383\times 10^{162} n_1+7.9031\times 10^{164}}\leq n_2\leq \text{3.5534074004528205$\grave{ }$*${}^{\wedge}$-79} \sqrt{-4.55384\times 10^{157} n_1^2+1.50383\times 10^{162} n_1+7.9031\times 10^{164}}+\text{1.7152185859604652$\grave{ }$*${}^{\wedge}$-56} \left(3.20659\times 10^{56} n_1+5.82404\times 10^{59}\right)\right)\right)$

In such cases, how can I find the maximum value of $n_1$ and $n_2$ from the Reduce output?

Edit: Please find below the function and the reduce operations that produce the two kind of outputs:

driftParamSet = (-0.72 
\!\(\*SubsuperscriptBox[\(n\), \(1\), \(2\)]\) - 
    Subscript[n, 
     1] (0.35` (0.8` - 0.39 Subscript[n, 2]) + 
       0.8` (-2.35 - 0.1` Subscript[n, 2])) - 
    0.19 Subscript[n, 
     2] (0.39` - 0.1` Subscript[n, 2] + 
       0.1` (-3 + 2 Subscript[n, 2])))/(0.8` Subscript[n, 1] + 
    0.19 Subscript[n, 2])

Reduce[driftParamSet > -5 && Subscript[n, 1] >= 0 && 
  Subscript[n, 2] >= 0 , {Subscript[n, 1], Subscript[n, 
  2]}, Integers]
Reduce[driftParamSet > -1000  && Subscript[n, 1] >= 0 && 
  Subscript[n, 2] >= 0 , {Subscript[n, 1], Subscript[n, 2]}, Integers]

Answer

You can use Maximize on the first example:

r1 = Quiet @ Reduce[
    driftParamSet > -5 && Subscript[n, 1] >= 0 && Subscript[n, 2] >= 0,
    {Subscript[n, 1], Subscript[n, 2]},

    Integers
];

Maximize[{Subscript[n, 2], r1}, {Subscript[n, 1], Subscript[n, 2]}]

{657., {Subscript[n, 1] -> 102., Subscript[n, 2] -> 657}}

For the second example, Maximize is unable to find a result, and then uses NMaximize:

r2 = Quiet @ Reduce[

    driftParamSet > -1000  && Subscript[n, 1] >= 0 && Subscript[n, 2] >= 0,
    {Subscript[n, 1], Subscript[n, 2]},
    Integers
];

Maximize[{Subscript[n, 2], r2}, {Subscript[n, 1], Subscript[n, 2]}]

NMaximize::cvmit: Failed to converge to the requested accuracy or precision within 100 iterations.

{22712., {Subscript[n, 1] -> 1118, Subscript[n, 2] -> 22712}}

As the error message says, 100 iterations were not sufficient. So, switch to using NMaximize, and raise the iteration maximum:

NMaximize[
    {Subscript[n, 2], r2},
    {Subscript[n, 1], Subscript[n, 2]},
    MaxIterations -> 2000
]

{114663., {Subscript[n, 1] -> 17793, Subscript[n, 2] -> 114663}}