
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 parametric equations of curve $C$ are $\begin{cases} x=\cos \alpha \\ y=m+\sin \alpha\end{cases}$ (where $\alpha$ is the parameter), and the parametric equations of line $l$ are $\begin{cases} x=1+ \frac { \sqrt {5}}{5}t \\ y=4+ \frac {2 \sqrt {5}}{5}t\end{cases}$ (where $t$ is the parameter),  
$(1)$ find the general equations of curve $C$ and line $l$;  
$(2)$ if line $l$ intersects curve $C$ at points $P$ and $Q$, and $|PQ|= \frac {4 \sqrt {5}}{5}$, find the value of the real number $m$.

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