```if:java
___Note for Java users:___ Due to type checking in Java, inputs and outputs are formated quite differently in this language. See the footnotes of the description.


```

You have the following lattice points with their corresponding coordinates and each one with an specific colour.

 ```
Point   [x ,  y]     Colour
----------------------------
  A     [ 3,  4]     Blue
  B     [-7, -1]     Red
  C     [ 7, -6]     Yellow
  D     [ 2,  5]     Yellow
  E     [ 1, -5]     Red
  F     [-1,  4]     Red
  G     [ 1,  7]     Red
  H     [-3,  5]     Red
  I     [-3, -5]     Blue
  J     [ 4,  1]     Blue
```
We want to count the triangles that have the three vertices with the same colour. The following picture shows the distribution of the points in the plane with the required triangles.

![source: imgur.com](http://i.imgur.com/sP0l1i1.png)

The input that we will have for the field of lattice points described above is:

```
[[[3, -4], "blue"],  [[-7, -1], "red"], [[7, -6], "yellow"], [[2, 5], "yellow"],
 [[1, -5], "red"],   [[-1, 4], "red"],  [[1, 7], "red"],     [[-3, 5], "red"], 
 [[-3, -5], "blue"], [[4, 1], "blue"] ]
```
We see the following result from it:

```
Colour   Amount of Triangles       Triangles
Yellow         0                    -------
Blue           1                      AIJ
Red            10                   BEF,BEG,BEH,BFG,BFH,BGH,EFG,EFH,EHG,FGH
```
As we have 5 different points in red and each combination of 3 points that are not aligned.

We need a code that may give us the following information in order:

```
1) Total given points
2) Total number of colours
3) Total number of possible triangles
4) and 5) The colour (or colours, sorted alphabetically) with the highest amount of triangles
```

In Python our function will work like:

```
[10, 3, 11, ["red",10]]) == count_col_triang([[[3, -4], "blue"],  [[-7, -1], "red"], [[7, -6], "yellow"], [[2, 5], "yellow"], 
                                              [[1, -5], "red"],   [[-1, 4], "red"],  [[1, 7], "red"],     [[-3, 5], "red"],
                                              [[-3, -5], "blue"], [[4, 1], "blue"] ])
```
In the following case we have some points that are aligned and we have less triangles that can be formed:

```
[10, 3, 7, ["red", 6]] == count_col_triang([[[3, -4], "blue"],  [[-7, -1], "red"], [[7, -6], "yellow"], [[2, 5], "yellow"],
                                            [[1, -5], "red"],   [[1, 1], "red"],   [[1, 7], "red"],     [[1, 4], "red"], 
                                            [[-3, -5], "blue"], [[4, 1], "blue"] ])
```
Just to see the change with the previous case we have this:

![source: imgur.com](http://i.imgur.com/cCgO7ql.png)

In the special case that the list of points does not generate an even single triangle, the output will be like this case:

```
[9, 3, 0, []] == count_col_triang([[[1, -2], "red"], [[7, -6], "yellow"], [[2, 5], "yellow"], [[1, -5], "red"],
                                  [[1, 1], "red"],   [[1, 7], "red"],     [[1, 4], "red"],    [[-3, -5], "blue"], 
                                  [[4, 1], "blue"] ])
```
It will be this case:

![source: imgur.com](http://i.imgur.com/VB7t7Ij.png)

If in the result we have two or more colours with the same maximum amount of triangles, the last list should be like (e.g)

```
[35, 6, 35, ["blue", "red", "yellow", 23]]     # having the names of the colours sorted alphabetically
```
For the condition of three algined points A, B, C, you should know that the the following determinant should be 0.

```
 | xA    yA    1|
 | xB    yB    1|    = 0
 | xC    yC    1|

```
Assumptions:

- In the list you have unique points, so a point can have only one colour.

- All the inputs are valid

Enjoy it!

````if:java
---

___For java users:___

Two immutable objects, `ColouredPoint` and `TriangleResult`, have been designed for you in the preloaded part. You will receive inputs as lists of ColouredPoint objects and will return a TriangleResult object. For the last one, you may note the organization of the arguments of the constructor which differs a bit from the description above.

You may find below the signatures of the available methods of these objects:

````

```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 = 'from itertools import combinations as combos\n\ndef count_col_triang(in_):\n    cl = list(set([i[1] for i in in_])) # cl is the full color-list\n    \n    d = {} # dictionary of color keys and triangle-count values\n    for c in cl: # i[0] + [1] is a row of the test matrix: [xi, yi, 1]\n        triples = combos([i[0] + [1] for i in in_ if i[1] == c], 3)\n        test_matrices = list(map(list, triples))\n        d[c] = len([i for i in test_matrices if determinant(i)])\n        \n    mc = max(d, key=d.get) # maximally-triangled color\n    maxcl = sorted([c for c in cl if d[c] and d[c] == d[mc]])\n    sumtri = sum([d[i] for i in d])\n    return [len(in_), len(cl), sumtri, maxcl + [d[mc]] if sumtri else []]\n\ndef determinant(m):\n    a, b, c, d, e, f, g, h, i = tuple(m[0] + m[1] + m[2])\n    return a*(e*i-f*h) - b*(d*i-f*g) + c*(d*h - e*g)'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'count_col_triang'
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: 9130