Here's a way to construct a list containing every positive rational number:

Build a binary tree where each node is a rational and the root is `1/1`, with the following rules for creating the nodes below:
* The value of the left-hand node below `a/b` is `a/a+b`
* The value of the right-hand node below `a/b` is `a+b/b`

So the tree will look like this:

```
                       1/1
                  /           \ 
            1/2                  2/1
           /    \              /     \
       1/3        3/2        2/3       3/1
      /   \      /   \      /   \     /   \
   1/4    4/3  3/5   5/2  2/5   5/3  3/4   4/1
 
 ...
```

Now traverse the tree, breadth first, to get a list of rationals.

```
[ 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, .. ]
```

Every positive rational will occur, in its reduced form, exactly once in the list, at a finite index.

```if:haskell
In the kata, we will use tuples of type `(Integer, Integer)` to represent rationals, where `(a, b)` represents `a / b`
```
```if:javascript
In the kata, we will use tuples of type `[ Number, Number ]` to represent rationals, where `[a,b]` represents `a / b`
```

Using this method you could create an infinite list of tuples:

matching the list described above:

However, constructing the actual list is too slow for our purposes. Instead, study the tree above, and write two functions:

For example:

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'def rat_at(n):\n    if n == 0:\n        return 1,1\n    # find line\n    line = 1\n    tot = 0\n    while tot <= n:\n        tot += 2**line\n        line += 1\n    if line > 1:\n        line -= 1\n    route = \'\'\n    b = 2**line # nr of elements in the line\n    pos = n+1 - 2**line # pos in the line\n    while b > 1:\n        if pos < b // 2:\n            route += \'l\'\n            b = b // 2\n            # pos does not change\n        else:\n            route += \'r\'\n            b = b // 2\n            pos -= b\n    a, b = 1, 1\n    for c in route:\n        if c == \'l\':\n            a, b = a, a+b\n        else:\n            a, b = a+b, b\n    return a,b\n\ndef index_of(a, b):\n    aa = a\n    bb = b\n    route = \'\'    \n    while not (aa == 1 and bb == 1):\n        if aa > bb:\n            aa = aa-bb\n            route = \'r\' + route \n        else:\n            bb = bb-aa\n            route = \'l\' + route\n    above = 0\n    for i in range(len(route)):\n        above += 2**i  \n    pos = [0,2**len(route)-1]\n    for c in route:\n        if c == \'l\':\n            pos[1] = pos[1] - (pos[1]+1 - pos[0]) // 2\n        else:\n            pos[0] = pos[0] + (pos[1]+1 - pos[0]) // 2\n    print(("a: {0}, b: {1}, above: {2}, pos: {3}, route: {4}".format(a,b, above, pos, route)))\n    return above+pos[0]\n    \n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'rat_at'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 5558