
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 \( P \) is a point on the right branch of the hyperbola \( C: \frac{x^2}{2} - y^2 = 1 \), line \( l \) is an asymptote of the hyperbola \( C \), and the projection of \( P \) onto \( l \) is \( Q \). Let \( F_1 \) be the left focal point of the hyperbola \( C \). Find the minimum value of \( |PF_1| + |PQ| \).

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