
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 \( t_1, t_2 \) be the two distinct real roots of the equation \( t^2 - (5a - 2)t - 3a^2 - 7a + 1 = 0 \). Find the value of the real number \( a \) such that, for any non-zero real number \( m \), the function \( f(x) = \cos (m \pi x) \cos \left[\left(t_{1}^{3} + t_{2}^{3}\right) \pi x\right] \) is a periodic function.

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