
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
What is the transformation matrix $\mathbf{M}$ that performs the transformation which sends rectangle $PQRS$ to rectangle $P'Q'R'S'$, where the rectangle rotates by $90^\circ$ counter-clockwise and scales by a factor of 3?

[asy]
size(200);
import graph;

pair Z=(0,0), P=(0,0), Q=(2,0), R=(2,1), S=(0,1);
pair P1 = (0,0), Q1=(-1,0), R1=(-1,3), S1=(0,3);

Label f; 
f.p=fontsize(6); 
xaxis(-3,3,Ticks(f, 1.0)); 
yaxis(-3,3,Ticks(f, 1.0));

draw(P--Q--R--S--cycle,red);
dot(P); label("$P=P'$", P, SE);
dot(Q); label("$Q$", Q, SE);
dot(R); label("$R$", R, NE);
dot(S); label("$S$", S, NW);

draw(P1--Q1--R1--S1--cycle, blue);
dot(Q1); label("$Q'$", Q1, SW);
dot(R1); label("$R'$", R1, NW);
dot(S1); label("$S'$", S1, N);

[/asy]

Present the answer in LaTex format: \boxed{Your answer}
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