
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 $\overrightarrow{a} = (1,2)$, $\overrightarrow{b} = (4,2)$, and $\overrightarrow{c} = m\overrightarrow{a} + \overrightarrow{b}$ ($m \in \mathbb{R}$), such that the angle between $\overrightarrow{c}$ and $\overrightarrow{a}$ is equal to the angle between $\overrightarrow{c}$ and $\overrightarrow{b}$, find the value of $m$.

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