
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 equation of the line $l$ as
$$
\begin{cases}
x=t \\
y=2t+1
\end{cases}
$$
where $t$ is the parameter, and the parametric equation of the circle $C$ as
$$
\begin{cases}
x=a\cos\theta \\
y=a\sin\theta
\end{cases}
$$
with $a>0$ and $\theta$ as the parameter. Let point $P$ be any point on circle $C$. If the maximum distance from point $P$ to line $l$ is $\frac{\sqrt{5}}{5}+1$, find the value of $a$.

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