
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 the plane vectors $\overrightarrow{a} = (2m - 1, 2)$ and $\overrightarrow{b} = (-2, 3m - 2)$, and $| \overrightarrow{a} + \overrightarrow{b} | = | \overrightarrow{a} - \overrightarrow{b} |$. Find the projection of $5 \overrightarrow{a} - 3 \overrightarrow{b}$ on the vector $\overrightarrow{a}$.

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