
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 point P is a moving point on the parabola $y^2 = 4x$, and point $A(0, 2\sqrt{2})$ is fixed. If a line PQ is drawn perpendicular to the y-axis, passing through point P and intersecting the y-axis at point Q, then the minimum value of $|PA| + |PQ|$ is \_\_\_\_\_\_.

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