
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
Let  $ABCD$  be a rectangle with  $AB=10$  and  $BC=26$ . Let  $\omega_1$  be the circle with diameter  $\overline{AB}$  and  $\omega_2$  be the circle with diameter  $\overline{CD}$ . Suppose  $\ell$  is a common internal tangent to  $\omega_1$  and  $\omega_2$  and that  $\ell$  intersects  $AD$  and  $BC$  at  $E$  and  $F$  respectively. What is  $EF$ ?

[asy]
        size(10cm);
        draw((0,0)--(26,0)--(26,10)--(0,10)--cycle);
        draw((1,0)--(25,10));
        draw(circle((0,5),5));
        draw(circle((26,5),5));
        dot((1,0));
        dot((25,10));
        label(" $E$ ",(1,0),SE);
        label(" $F$ ",(25,10),NW);
        label(" $A$ ", (0,0), SW);
        label(" $B$ ", (0,10), NW);
        label(" $C$ ", (26,10), NE);
        label(" $D$ ", (26,0), SE);
        dot((0,0));
        dot((0,10));
        dot((26,0));
        dot((26,10));
[/asy]

*Proposed by Nathan Xiong*

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