
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 non-zero vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy: $(|\overrightarrow{a}| = |\overrightarrow{b}| = |\overrightarrow{a} + \overrightarrow{b}|)$, $(( \overrightarrow{a} + \overrightarrow{b}) \perp (2 \overrightarrow{a} + \lambda \overrightarrow{b}))$, find the value of real number $\lambda$.

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