
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 right triangle $DEF$, $\cos{E}=\frac{5\sqrt{221}}{221}$. Find $DF$ where $EF$ is the hypotenuse and its length is $\sqrt{221}$.

[asy]
draw((0,0)--(7,0)--(0,9)--cycle,black+linewidth(1));
draw(rightanglemark((7,0),(0,0),(0,9),20),black+linewidth(1));
label("$D$",(0,0),W);
label("$F$",(7,0),E);
label("$E$",(0,9),W);
label("$\sqrt{221}$",(7,0)--(0,9),NE);
[/asy]

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