
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 $C$: $\frac{x^2}{2} - y^2 = 1$, and the line $l$ is an asymptote of the hyperbola $C$. The projection of $P$ on $l$ is $Q$, and $F_1$ is the left focus of the hyperbola $C$. Calculate the minimum value of $|PF_1| + |PQ|$.

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