
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
(1) Given vectors $\overrightarrow{a} = (1, 2)$, $\overrightarrow{b} = (-3, 2)$, $\overrightarrow{c} = (3, 4)$. If $\lambda$ is a real number, and $(\overrightarrow{a} + \lambda\overrightarrow{b})$ is parallel to $\overrightarrow{c}$, find the value of $\lambda$.
(2) Given non-zero vectors $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ that are not collinear. If vectors $k\overrightarrow{e_{1}} + \overrightarrow{e_{2}}$ and $\overrightarrow{e_{1}} + k\overrightarrow{e_{2}}$ are to be collinear, determine the real number $k$.

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