
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 parabola $C_{1}$: $y^{2}=x$ and the circle $C_{2}$: $(x-3)^{2}+y^{2}=2$.<br/>$(1)$ If the directrix of the parabola $C_{1}$ intersects the $x$-axis at point $T$, and $MN$ is a chord passing through the focus $F$ of $C_{1}$, find the minimum value of $\overrightarrow{TM}\cdot \overrightarrow{TN}$;<br/>$(2)$ Given that $P$, $A$, $B$ are three distinct points on the parabola $C_{1}$, and point $P$ is not the origin. If the chords $PA$ and $PB$ cut the circle $C_{2}$ into segments of length $2$ each, and $PA=PB$, find the coordinates of point $P$.

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