I remember that long time ago, at a math conference it was presented a Maple package on Functional Equations.
Here is a recent paper of one of the authors:

https://link.springer.com/article/10.1007/s00010-023-00978-7

P.S. By hand, or via AI it is easy to solve it: by a substitution it results a Cauchy functional equation.

@dharr Nice work and imagination. Vote up!

@dharr This works only for very "similar" sequences. Trying with (1/n) and (1/2^n) we immediadely get a FAIL result. 

@Alfred_F But you will not be able to avoid  (-1)^n or cos(n*Pi) or ...

u := n -> s((n+1)/2) * (1 - (-1)^n)/2 + t(n/2) * (1 + (-1)^n)/2;

Probably AI will know to find such one if you ask it to use (-1)^n.

@Alfred_F But  

u:=n -> `if`(n::odd, s((n+1)/2), t(n/2)):

is one formula. It can by changed into a form containing (-1)^n instead of the function `if`, but why would you prefer a more complicated version?

@Kitonum Dividing by 10, you have assumed that x,y,z are multiple of 10, making the problem much easier! By pure luck such a solution exists.

@Alfred_F You did not mention any condition. Do you know an example where continuity is necessary?

[With my assumption,  the existence of at least one planar section with more than 1 point is clear because a point A can be chosen outside of the tangent plane. And the Gateaux differentiability (hence continuity) is needed in a single point].

I think the fact that the former Maple UI was better and faster can be proved mathematically :-)

@Alfred_F  To factor, just use:

evalindets(%, `^`, factor);

                      81*(x-5)^4 + 117649*(y-11)^6 + 256*(z-13)^8

@C_R 

    The term residual for an antiderivative does not make sense.
    The constant depends on the method of integration, even in the simplest cases.
    For example:
 ∫(x+1)dx = 1/2(x+1)²   if we use the formula ∫(ax+b)ⁿdx=(1/a)(((ax+b)ⁿ⁺¹)/(n+1))
    but
∫(x+1)dx = 1/2x²+x   if we integrate term by term.

@C_R 

int(D(tanh)(x), x);
diff(%, x);
simplify(%) = D(tanh)(x);  ###
is(%);

@nm  For implicitplot,  rhs(sol) is not needed.

@dharr  This problem is known and treated in Odrzywolek' paper. It can be also solved by changing the argument function from (-Pi,Pi] to [-Pi,Pi). Anyway, it's a game ...

A good idea for Maplesoft would be to enhance the Maple Kernel for Jupyter. It works ok, but some useful elements of the GUI (e.g. the animations) are still missing.