
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
The parametric equation of curve $C_{1}$ is: 
$$
\begin{cases}
x=1+t\cos \frac{\pi}{4} \\
y=5+t\sin \frac{\pi}{4}
\end{cases} \quad (t \text{ is the parameter})
$$
The parametric equation of curve $C_{2}$ is: 
$$
\begin{cases}
x=\cos \phi \\
y= \sqrt{3}\sin \phi
\end{cases} \quad (\phi \text{ is the parameter})
$$
$(1)$ Find the general equation of curve $C_{2}$. If the coordinate origin is taken as the pole and the positive half-axis of $x$ as the polar axis to establish the coordinate system, find the polar coordinate equation of curve $C_{1}$;  
$(2)$ If point $P$ is any point on curve $C_{2}$, find the minimum distance from point $P$ to curve $C_{1}$.

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