Quadratic Function Operations and Queries

You are given a quadratic function in the form $$y=ax^2+bx+c,$$ where \(a\neq 0\). Initially, you may be asked to compute its extreme value (minimum if \(a>0\) or maximum if \(a<0\)). However, the difficulty is increased by a series of operations that modify the function or query its properties.

The following operations can be applied sequentially:

1. Vertical Shift: Given a coefficient \(k\), shift the function vertically. That is, update the function as $$y = ax^2 + bx + (c+k).$$
2. Horizontal Shift: Given a coefficient \(k\), shift the function horizontally. The new function becomes $$y = a(x-k)^2 + b(x-k) + c.$$ After expanding, the coefficients update to: $$a'=a,\quad b' = b-2ak,\quad c' = ak^2 - bk + c.$$
3. Reflection about a Point: Given coefficients \(k_1\) and \(k_2\), reflect the function about the point \((k_1,k_2)\). The new function is given by $$y = 2k_2 - f(2k_1 - x).$$ For \(f(x)=ax^2+bx+c\), after expanding, the updated coefficients are: $$a'=-a,\quad b'=4ak_1+b,\quad c'=2k_2-(4ak_1^2+2bk_1+c).$$
4. Interval Extreme Query: Given two numbers \(k_1\) and \(k_2\) (with \(k_1\le k_2\)), compute the minimum and maximum values of the current quadratic function on the closed interval \([k_1,k_2]\). Recall that for a quadratic function \(f(x)=ax^2+bx+c\), the vertex is at \(x=-\frac{b}{2a}\). Evaluate \(f(k_1)\), \(f(k_2)\), and \(f(-\frac{b}{2a})\) (if the vertex lies in \([k_1,k_2]\)) to determine the extreme values.
5. Intersection Query: Given coefficients \(u,v,w\) defining another quadratic function $$y_2=ux^2+vx+w,$$ determine whether the current quadratic function \(y=ax^2+bx+c\) and \(y_2\) have any intersection points.
   Hint: They intersect if the equation $$(a-u)x^2+(b-v)x+(c-w)=0$$ has at least one real solution. Be careful with the case when \(a-u=0\).

After processing all operations, output the extreme value of the final quadratic function. Specifically, output the minimum value if \(a>0\) or the maximum value if \(a<0\) (which is achieved at the vertex \(x=-\frac{b}{2a}\)).

Input Format:

• The first line contains three integers \(a\), \(b\), and \(c\) representing the initial quadratic function \(y=ax^2+bx+c\).
• The second line contains an integer \(Q\) indicating the number of operations.
• The following \(Q\) lines each describe an operation. Each operation starts with an integer indicating its type:
  • Type 1: 1 k
  • Type 2: 2 k
  • Type 3: 3 k1 k2
  • Type 4: 4 k1 k2 — perform a query on the interval \([k1,k2]\) and output the minimum and maximum values (separated by a space).
  • Type 5: 5 u v w — perform a query to check whether the current function and the quadratic function \(y_2=ux^2+vx+w\) have an intersection. Output YES if they intersect, and NO otherwise.

Output Format:

• For each operation of type 4, output a line containing two numbers: the minimum and maximum values of the quadratic function on the given interval.
• For each operation of type 5, output a line containing either YES or NO depending on whether an intersection exists.
• After processing all operations, output a final line containing the extreme value (minimum if \(a>0\) or maximum if \(a<0\)) of the final quadratic function.

It is guaranteed that the input values are such that all operations can be performed without ambiguity.

inputFormat

The input begins with a line containing three integers \(a\), \(b\), and \(c\) (with \(a\neq0\)), representing the quadratic function \(y=ax^2+bx+c\). The next line contains an integer \(Q\) representing the number of operations. The following \(Q\) lines each describe an operation in one of the following formats:

• 1 k
• 2 k
• 3 k1 k2
• 4 k1 k2
• 5 u v w

For operation type 4, it is guaranteed that \(k1 \le k2\).

outputFormat

For each operation of type 4 or 5, output a line with the corresponding result. For type 4, print the minimum value followed by a space and the maximum value on the specified interval. For type 5, print YES if the current quadratic function intersects with the given quadratic function, otherwise print NO. Finally, after all operations have been processed, output one line containing the extreme value (minimum if \(a>0\) or maximum if \(a<0\)) of the final quadratic function.

sample

1 -2 1
3
1 3
4 0 2
5 1 -2 1

3 4
NO
3

</p>