
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
For \( \mathbf{v} = \begin{pmatrix} 2 \\ y \end{pmatrix} \) and \( \mathbf{w} = \begin{pmatrix} 5 \\ 10 \end{pmatrix} \),
\[
\text{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 10 \\ 20 \end{pmatrix}.
\]
Find \( y \).

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