
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, triangles $ABC$ and $BDE$ are isosceles. The perimeter of $\triangle BDE$ is $26,$ the perimeter of $\triangle ABC$ is $24,$ and the length of $BE$ is $10.$ What is the length of $AB?$

[asy]
size(7cm);
defaultpen(fontsize(11));
pair b = (0, 0);
pair e = 10 * dir(-30);
pair a = 8 * dir(-140);
pair d = 6 * dir(-90);
draw(a--b--e--d--cycle);
draw(b--d);
label("$y^\circ$", a, 2 * (E + NE));
label("$y^\circ$", b, 2 * (S + SW));
label("$x^\circ$", b, 2 * (S + SE));
label("$x^\circ$", e, 2 * (2 * W));
label("$A$", a, W);
label("$B$", b, N);
label("$E$", e, E);
label("$D$", d, S);
[/asy]

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