
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
Given that $F$ is the focus of the curve $\begin{cases} x=2\sqrt{2}\cos \theta \\ y=1+\cos 2\theta \end{cases} (\theta \in \mathbb{R})$, and $A(1,0)$, then the value of $|AF|$ equals \_\_\_\_\_\_.

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