
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 vectors $\overrightarrow{a}=(2,1)$, $\overrightarrow{b}=(1,1)$, $\overrightarrow{c}=(5,2)$, $\overrightarrow{m}=\lambda \overrightarrow{b}+ \overrightarrow{c}$ (where $\lambda$ is a constant).  
$(1)$ Find $\overrightarrow{a}+ \overrightarrow{b}$;  
$(2)$ If $\overrightarrow{a}$ is parallel to $\overrightarrow{m}$, find the value of the real number $\lambda$.

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