
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
In the diagram below, lines $k$ and $\ell$ meet at a point. Line $m$ is drawn such that it intersects line $k$ at a $45^\circ$ angle and line $\ell$ at a $30^\circ$ angle. Find the measure of angle $x$, the angle formed at the intersection on line $m$ with line $\ell$.

[asy]
size(200);
import markers;
pair A = dir(-22)*(0,0);
pair B = dir(-22)*(4,0);
pair C = dir(-22)*(4,-1.5); // New angle intersection point lower
pair D = dir(-22)*(0,2);
pair F = dir(-22)*(0,1);
pair G = dir(-22)*(4,1);
pair H = dir(-22)*(2,1); 

markangle(Label("$45^\circ$",Relative(0.4)),n=1,radius=15,F,H,A); 
markangle(Label("$x$",Relative(0.7)),n=1,radius=15,F,H,C); 
markangle(Label("$30^\circ$",Relative(0.6)),n=1,radius=20,C,H,D); 

pair X,Y;

X=A;
Y=B;
draw(X--Y);

X=A;
Y=C;
draw(X--Y);

X=C;
Y=B;
draw(X--Y);

X=B;
Y=D;
draw(X--Y);

X=G;
Y=F;
draw(X--Y);

label("$\ell$",1.4*A-.4*B);
label("$k$",1.4*F-.4*G);
[/asy]

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