
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 left focus of the hyperbola $\frac{x^2}{4} - \frac{y^2}{12} = 1$, $A(1,4)$, and $P$ is a moving point on the right branch of the hyperbola, find the minimum value of $|PF| + |PA|$.

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