Problem: Your task is the exact same as for the easy version. But this time, the marmots subtract the village's population P from their random number before responding to Heidi's request.

Also, there are now villages with as few as a single inhabitant, meaning that <image>.

Can you help Heidi find out whether a village follows a Poisson or a uniform distribution?

Input

Same as for the easy and medium versions. But remember that now 1 ≤ P ≤ 1000 and that the marmots may provide positive as well as negative integers.

Output

Output one line per village, in the same order as provided in the input. The village's line shall state poisson if the village's distribution is of the Poisson type, and uniform if the answers came from a uniform distribution.

Solution (in CPP):

Solution:
#pragma GCC optimize ("O3")
#pragma GCC target ("sse4")

#include <bits/stdc++.h>
#include <ext/pb_ds/tree_policy.hpp>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/rope>

using namespace std;
using namespace __gnu_pbds;
using namespace __gnu_cxx;
 
typedef long long ll;
typedef long double ld;
typedef complex<ld> cd;

typedef pair<int, int> pi;
typedef pair<ll,ll> pl;
typedef pair<ld,ld> pd;

typedef vector<int> vi;
typedef vector<ld> vd;
typedef vector<ll> vl;
typedef vector<pi> vpi;
typedef vector<pl> vpl;
typedef vector<cd> vcd;

template <class T> using Tree = tree<T, null_type, less<T>, rb_tree_tag,tree_order_statistics_node_update>;

#define FOR(i, a, b) for (int i=a; i<(b); i++)
#define F0R(i, a) for (int i=0; i<(a); i++)
#define FORd(i,a,b) for (int i = (b)-1; i >= a; i--)
#define F0Rd(i,a) for (int i = (a)-1; i >= 0; i--)

#define sz(x) (int)(x).size()
#define mp make_pair
#define pb push_back
#define f first
#define s second
#define lb lower_bound
#define ub upper_bound
#define all(x) x.begin(), x.end()

const int MOD = 1000000007;
const ll INF = 1e18;
const int MX = 3005;

int V;
ld prob0[1001][MX], prob1[1001][MX];

void gen(int mean) {
    ld a = 1;
    int divi = mean;
    F0R(i,MX) {
        prob0[mean][i] = a*exp(-divi);
        a *= mean; a /= i+1;
        while (divi && a/exp(1) >= 1) divi --, a /= exp(1);
    }
    F0R(i,MX) {
        if (prob0[mean][i] == 0) prob0[mean][i] = -INF;
        else prob0[mean][i] = log(prob0[mean][i]);
    }
    F0R(i,MX) prob1[mean][i] = -INF;
    F0R(i,2*mean+1) prob1[mean][i] = log((ld)1/(2*mean+1));
}

vi v;
ld a,b;

void test(int mean) {
    ld A = 0, B = 0;
    for (int i: v) {
        if (i+mean < 0) A = B = -INF;
        else if (i+mean >= MX) continue;
        else {
            A += prob0[mean][mean+i];
            B += prob1[mean][mean+i];
        }
    }
    a = max(a,A), b = max(b,B);
}

void solve() {
    v = {};
    F0R(i,250) { 
        // int x = rand() % 1501-750;
        int x; cin >> x;
        if (x > -MX && x < MX) v.pb(x); 
    }
    a = b = -INF;
    FOR(mean,1,1001) test(mean);
    if (a < b) cout << "uniform";
    else cout << "poisson";
    // cout << a << " " << b << "\n";
    cout << "\n";
}

int main() {
    ios_base::sync_with_stdio(0); cin.tie(0);
    FOR(i,1,1001) gen(i);
    /*F0R(i,MX) cout << prob0[1000][i] << "\n";
    exit(0);*/
    cin >> V;
    F0R(i,V) solve();
}

/* Look for:
* the exact constraints (multiple sets are too slow for n=10^6 :( ) 
* special cases (n=1?)
* overflow (ll vs int?)
* array bounds
* if you have no idea just guess the appropriate well-known algo instead of doing nothing :/
*/