I have the following code but recieve timeouts errors:

Source Code

def findPositions$($a$,$ b$,$ s$,$ queries$)$:

$"""$

Finds the positions based on the given queries.

Parameters:

a $($int$)$: The coefficient for updating x in the formulas.

b $($int$)$: The coefficient for updating y in the formulas.

s $($int$)$: The modulus value.

queries $($list$)$: A list of queries, where each query is represented as a list $[$x$,$ y$,$ n$],$

where x and y are initial positions and n is the number of iterations.

Returns:

list: A list of positions after performing the given number of iterations for each query.

$"""$

results $=[]$

for query in queries:

x$,$ y$,$ n $=$ query

for $\_$ in range$($n$)$:

# Update x and y according to the given formulas

x$\_$new $=($a$*$x $-$ y$)\%$ s

y$\_$new $=($x $+$ b$*$y$)\%$ s

# Prepare x and y for the next iteration

x$,$ y $=$ x$\_$new, y$\_$new

# Append the result for the current query

results.append$([$x$,$ y$])$

return results

if $\_\_$name$\_\_==\text{'}\_\_$main$\_\_\text{'}$:

# User input for a$,$ b$,$ and s

a$,$ b$,$ s $=$ map$($int$,$ input$("$Enter a$,$ b$,$ and s separated by spaces: $").$rstrip$().$split$())$

# Number of queries

q $=$ int$($input$("$Enter the number of queries: $").$strip$())$

# Queries input

queries $=[]$

print$("$Enter each query in the format $\text{'}$x y n$\text{'},$ each on a new line:"$)$

for $\_$ in range$($q$)$:

queries.append$($list$($map$($int$,$ input$().$rstrip$().$split$())))$

# Find positions based on queries

results $=$ findPositions$($a$,$ b$,$ s$,$ queries$)$

# Output results

print$("$Output:$")$

for result in results:

print$($f$\text{'}\{$result$[0]\}\{$result$[1]\}\text{'})$

Can you review the problems, specifically the constraint and give a more efficient solution

In a certain video game, the screen is square of a given size, $s,$ measured in pixels. The $x-$ and $y-$coordinates

go from $(0,0)$ to $(s-1,s-1).$ Boulders are rolling across the screen, and the player must avoid being hit

by them. When a boulder rolls off one edge of the screen, it appears at the opposite edge of the screen,

preserving the other coordinate. For example, if $s=100$ and a boulder on the middle of the right edge of the

screen, at coordinate $(99,50),$ moves to the right, it would reappear at coordinates $(0,50)($i$.$e$.$ in the middle on

the left edge of the screen$).$ The screen is essentially the surface of a torus $($doughnut shape$)$ where you can

imagine the right side has been joined to the left side, and the top of the screen to the bottom.

The rolling boulders are simulated by updating their $(x,y)$ coordinates in discrete steps. Each rolling boulder is

considered to move along a trajectory defined by a sequence of discrete points:

$(x_0,y_0),(x_1,y_1),$dots,$(x_i,y_i).$ It moves along a straight line joining consecutive points in the sequence. The

trajectory of a rolling boulder is parameterised by two integers a and $b$ so that $(x_{i+1},y_{i+1})$ is given by:

$x_{i+1}=a{x}_i-y_i$

$y_{i+1}=x_i+b{y}_i$

If either coordinate returned by the calculation is either negative or larger than $s,$ then it is considered to wrap

around to the other side of the screen, and carry on counting. For example, if $s=100,$ then the coordinate

$(131,257)$ would correspond to the point at $(31,57),$ and the coordinate $(-1,-3)$ would correspond to the

coordinate $(99,97).$Given the parameters $a,b$ and $s($which govern all of the boulders on the screen at the same time$),$ you are

asked to determine the result of each of $q$ queries. Each query is of the form $(x,y,n),$ and is asking for the

current position of a boulder that would have started at position $(x,y)$ and taken $n$ update steps.

Input Format

Line $1$: a b s

Line $2$: q

Each of the following $q$ lines is of the format:

$x$ y $n$

Constralnts

$(x,y)1a1000$

$1b1000$

$100s1\mathrm{.}5{10}^9$

$1q{10}^5$

$0x$

Output Format

For each query, output $on$ a separate line, the $(x,y)$ coordinates $of$ the result, separated $by$ a single space.