Range of Power Sum Equivalence

Reimu obtained three real numbers \( n\), \( a\), \( b\) satisfying \(4\le n\le 5\) and \(5\le a,b\le 10\). She then computed
\[ f(n)=n^a+n^b, \]
and the calculator displayed only the integer part of \( f(n) \), i.e. \(\lfloor n^a+n^b\rfloor\).

She wonders: if there exists a real number \( n'\) (with \( n'\ge0 \)) such that the calculator display of
\[ {n'}^a+{n'}^b \]
is exactly the same as that of \( n^a+n^b\) (that is, their integer parts are equal), what is the range of all possible \( n' \)? Formally, if we define the set
\[ S=\{n'\ge0\;:\;\lfloor {n'}^a+{n'}^b\rfloor = \lfloor n^a+n^b\rfloor\}, \]
find the value of
\[ \Delta = \sup S - \inf S. \]

Note that the function
\[ f(x)=x^a+x^b \]
is strictly increasing for \( x\ge0 \). Therefore, if we let
\[ T = \lfloor n^a+n^b \rfloor, \] then the condition is equivalent to finding the unique numbers \( L \) and \( U \) satisfying
\[ f(L)=T \quad \text{and} \quad f(U)=T+1. \] The answer for each query is \( s = U - L \).

If you are not sure how to compute \( n^k \), you may use the pow() function from the cmath library, where pow(n,k) computes \( n^k \) (in your language of choice).

Furthermore, there are \( T \) independent queries given in the input. For the \( i\)-th query you are given \( n_i, a_i, b_i \). Let the answer for the \( i\)-th query be \( s_i \); you are to output the sum
\[ S = \sum_{i=1}^{T} s_i. \]
The answer is accepted if its absolute error does not exceed \(10^{-2}\).

Note on Input Generation:
To reduce input/output, the queries are generated via a pseudo‐random process. (See the provided code snippet in the statement.) For the case when a parameter op=1, we have \( a=b \); otherwise, no special constraint is imposed.

inputFormat

The input begins with an integer \( T \) representing the number of queries. Each of the following \( T \) lines contains three real numbers \( n, a, b \) separated by spaces.

Constraints:
• \(4 \le n \le 5\)
• \(5 \le a, b \le 10\)

outputFormat

Output a single number: the sum of all ranges \( s_i \) for the queries, i.e. \(\sum_{i=1}^{T} s_i\). The answer is accepted if its absolute error is within \(10^{-2}\).

sample

1
4.0 5.0 5.0

0.00039063