The following task is a significantly harder version of task Words from the third stage of 16th Polish OI. It wasn't used in the contest itself, but is an extension for those who solved "Words" and want more. :-)

Let $h$ be a function acting on strings composed of the digits 0 and 1. The function h transforms the string w by replacing (independently and concurrently) every digit 0 with 1 and every digit 1 with the string ,,10”. For example $h(“\texttt{1001}”)=“\texttt{101110}”$, $h(“”)=“”$ (i.e. $h$ assigns an empty string to the empty string). Note that $h$ is an injection, or a one-to-one function. By $h_k$ we denote the function $h$ composed with itself $k$ times. In particular, $h_0$ is the identity function h0(w)=w.

We are interested in the strings of the form $h_k(“\texttt{0}”)$ for $k=0,1,2,3,…$ This sequence begins with the following strings:

$$“\texttt{0}”, “\texttt{1}”, “\texttt{10}”, “\texttt{101}”, “\texttt{10110}”, “\texttt{10110101}”$$

We call the string $x$ a substring of the string $y$ if it occurs in $y$ as a contiguous (i.e. one-block) subsequence. A sequence of integers $k_1,k_2,…,k_n$ is given. Your task is to check whether a string of the form $h_{k_1}(“\texttt{0}”)⋅h_{k_2}(“\texttt{0}”)⋅⋅⋅h_{k_n}(“\texttt{0}”)$ is a substring of $h_m(“\texttt{0}”)$ for some $m$, and if it is, you shuold find minimal such $m$.

Input

The first line of the standard input contains a single integer $t$, $1 ≤ t ≤ 13$, denoting the number of test units. The first line of each test unit's description contains one integer $n$, $1 ≤ n ≤ 100\,000$. The second line of each description holds $n$ non-negative integers $k_1,k_2,…,k_n$, separated by single spaces. The sum of the numbers in the second line of any test unit description does not exceed $10\,000\,000$.

Output

Your programme should print out t lines to the standard output, one for each test unit. Each line corresponding to a test unit should contain one word: TAK (yes in Polish - if $h_{k_1}(“\texttt{0}”)⋅h_{k_2}(“\texttt{0}”)⋅⋅⋅h_{k_n}(“\texttt{0}”)$ is a substring of $h_m(“\texttt{0}”)$ for some in that test unit, or NIE (no in Polish) otherwise.

Example

Input

2
2
1 2
2
2 0

Output

TAK
NIE

The string from the first test case is $“\texttt{110}”$ - it is a substring of $h_4(“\texttt{0}”)=“\texttt{10110}”$. In the second test unit there is a string $“\texttt{100}”$, which is not a substring of $h_m(“\texttt{0}”)$ for any $m$.