
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
Let $\overrightarrow{a} = (-1, 1)$, $\overrightarrow{b} = (x, 3)$, $\overrightarrow{c} = (5, y)$, $\overrightarrow{d} = (8, 6)$, and suppose $\overrightarrow{b} \parallel \overrightarrow{d}$, and $(4 \overrightarrow{a} + \overrightarrow{d}) \perp \overrightarrow{c}$.
(1) Find $\overrightarrow{b}$ and $\overrightarrow{c}$.
(2) Find the projection of $\overrightarrow{c}$ in the direction of $\overrightarrow{a}$.
(3) Find $\lambda_1$ and $\lambda_2$ such that $\overrightarrow{c} = \lambda_1 \overrightarrow{a} + \lambda_2 \overrightarrow{b}$.

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