
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
Let the parametric equation of line $l_1$ be $\begin{cases}x=1+t \\ y=a+3t\end{cases}$ (where $t$ is the parameter), and a polar coordinate system is established with the origin as the pole and the positive $x$-axis as the polar axis, resulting in the equation of line $l_2$ being $\rho\sin \theta-3\rho\cos \theta+4=0$. If the distance between line $l_1$ and $l_2$ is $\sqrt{10}$, then the value of the real number $a$ is ______.

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