
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
A gate consists of two parts hinged at the top where each part is supported by a triangular frame. The triangular frames are arranged as: [asy]
import olympiad;
import math;

// Draw triangles
pair A = (0, 2);
pair B = (-1, 0);
pair C = (1, 0);
pair D = (3, 2);
pair E = (2, 0);
pair F = (4, 0);
draw(A--B--C--cycle);
draw(D--E--F--cycle);
draw(A--D);
label('$A$',A,N);
label('$B$',B,S);
label('$C$',C,S);
label('$D$',D,N);
label('$E$',E,S);
label('$F$',F,S);
[/asy]
If $\triangle ABC$ and $\triangle DEF$ have $\angle BAC = 25^\circ$ and $\angle EDF = 40^\circ$, and given the triangles are isosceles with $AB = AC$ and $DE = DF$, what is the sum $\angle BAC + \angle EDF$?

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