
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
In the Cartesian coordinate system, the graph of a linear function $y=kx+1\left(k\neq 0\right)$ passes through the point $A\left(1,3\right)$.  
$(1)$ Find the value of $k$;  
$(2)$ The analytical expression of the function obtained by shifting the graph of this linear function up by $2$ units is ______.

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