
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 the ellipse $C: \frac{x^{2}}{3}+y^{2}=1$, with $F_1$ and $F_2$ as its left and right foci, respectively.
(1) If point $A(\sqrt{2},2)$, and $P$ is any point on the ellipse, find the maximum value of $|PA|+|PF_1|$.
(2) The line $y=kx+\sqrt{2}$ intersects the trajectory of point $Q$ at two distinct points $A$ and $B$, and $\overrightarrow{OA} \cdot \overrightarrow{OB}=1$ (where $O$ is the origin), find the value of $k$.

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