
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 circle centered at $P$ has radius 2 and contains the point $D$. Segment $DE$ is tangent to the circle at $D$ and $\angle
DPE=\phi$. If point $F$ lies on $\overline{PD}$ and $\overline{EF}$ bisects $\angle DPE$, then express $PF$ in terms of $u$ and $v,$ where $u = \sin \phi$ and $v = \cos \phi.$

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