
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 rectangle $JKLM$, $P$ is a point on $LM$ so that $\angle MPK=90^\circ$. $UV$ is perpendicular to $LM$ with $LU=UP$, as shown.  $PK$ intersects $UV$ at $Q$.  Point $R$ is on $KM$ such that $RK$ passes through $Q$.  In $\triangle PQK$, $PK=30$, $KQ=18$ and $PQ=24$. Find $MV$. (Express your answer as a common fraction.) [asy]
size(7cm);defaultpen(fontsize(9));
real mv = 7/9 * 18;
path extend(pair a, pair b) {return a--(10 * (b - a));}

// Rectangle
pair j = (0, 0); pair l = (0, 24); pair m = (30 + mv, 0); pair k = (m.x, l.y);
draw(j--l--k--m--cycle);

// Extra points and lines
pair q = (30, 10.5); pair u = (q.x, 0); pair v = (q.x, l.y);
pair r = IP(k--m, extend(j, q));
pair p = (18, l.y);
draw(q--j--p--k--r--cycle);draw(v--u);
label("$R$", r, E); label("$P$", p, N);label("$Q$", q, 1.2 * NE + 0.2 * N);label("$U$", u, S); label("$V$", v, N);

// Right angles and tick marks
markscalefactor = 0.1;
draw(rightanglemark(j, l, p)); draw(rightanglemark(p, v, u)); draw(rightanglemark(q, u, m));draw(rightanglemark(j, p, q));
add(pathticks(l--p, 2, spacing=3.4, s=10));add(pathticks(p--v, 2, spacing=3.5, s=10));

// Number labels
label("$24$", midpoint(j--l), W); label("$30$", midpoint(j--p), NW); label("$24$", midpoint(p--q), NE);
label("$18$", midpoint(j--q), 0.8 * S + E);
[/asy]

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