
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 vectors \(\mathbf{u}\) and \(\mathbf{v}\) such that \(\|\mathbf{u}\| = 5\), \(\|\mathbf{v}\| = 10\), and \(\|\mathbf{u} + \mathbf{v}\| = 12\). Find \(\cos \phi\), where \(\phi\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\).

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