
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 trapezoid $EFGH$, the sides $EF$ and $GH$ are equal. The length of $FG$ is 10 and the length of $EH$ is 20. Each of the non-parallel sides makes right triangles with legs 5 and an unknown base segment. What is the perimeter of $EFGH$?
[asy]
pen p = linetype("4 4");
draw((0,0)--(4,5)--(12,5)--(16,0)--cycle);
draw((4,0)--(4,5), p);
draw((3.5,0)--(3.5, 1)--(4.0,1));

label(scale(0.75)*"E", (0,0), W);
label(scale(0.75)*"F", (4,5), NW);
label(scale(0.75)*"G", (12, 5), NE);
label(scale(0.75)*"H", (16, 0), E);
label(scale(0.75)*"10", (8,5), N);
label(scale(0.75)*"20", (8,0), S);
label(scale(0.75)*"5", (4, 2.5), E);
[/asy]

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