
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 a complex number $z$ satisfies $(1+i)z=1-3i$ (where $i$ is the imaginary unit),  
(1) Find the imaginary part of the complex number $z$;  
(2) If the complex number $(1+ai)z$ is a pure imaginary number, find the value of the real number $a$;  
(3) If the conjugate of the complex number $z$ is denoted as $\overset{ .}{z}$, find the modulus of the complex number $\frac { \overset{ .}{z}}{z+1}$.

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