
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 \(\mathbf{w}_0\) be a vector. The vector \(\mathbf{w}_0\) is projected onto \(\begin{pmatrix} 4 \\ 2 \end{pmatrix}\), resulting in the vector \(\mathbf{w}_1\). The vector \(\mathbf{w}_1\) is then projected onto \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\), resulting in the vector \(\mathbf{w}_2\). Find the matrix that takes \(\mathbf{w}_0\) to \(\mathbf{w}_2\).

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