
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 square $ABCD$, points $K$ and $L$ lie on $\overline{AB}$ and $\overline{DA}$, respectively, so that $AK=AL$. Points $M$ and $N$ lie on $\overline{BC}$ and $\overline{CD}$, respectively, and points $P$ and $Q$ lie on $\overline{KL}$ so that $\overline{MP} \perp \overline{KL}$ and $\overline{NQ} \perp \overline{KL}$. Triangle $AKL$, quadrilateral $BMKP$, quadrilateral $DLNQ$, and pentagon $MCNQP$ each has area $2$. What is $MP^2$?
[asy] real x=2sqrt(2); real y=2sqrt(16-8sqrt(2))-4+2sqrt(2); real z=2sqrt(8-4sqrt(2)); pair A, B, C, D, K, L, M, N, P, Q; A = (0,0); B = (4,0); C = (4,4); D = (0,4); K = (x,0); L = (0,x); M = (4,y); N = (y,4); P = M + z * dir(225); Q = N + z * dir(225); draw(A--B--C--D--A); draw(L--K); draw(Q--N^^M--P); draw(rightanglemark(N, Q, P), linewidth(.5)); draw(rightanglemark(M, P, K), linewidth(.5)); dot("$A$", A, S); dot("$B$", B, S); dot("$C$", C, dir(90)); dot("$D$", D, dir(90)); dot("$K$", K, S); dot("$L$", L, W); dot("$M$", M, dir(0)); dot("$N$", N, N); dot("$P$", P, SW); dot("$Q$", Q, SW); [/asy]

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