
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 $f(x)= \begin{cases} 2^{x-1} & (x\geqslant 1) \\ 3x-2 & (x < 1) \end{cases}$, the minimum integer value of $\lambda$ such that the inequality $f(\cos ^{2}\theta+ \lambda\sin \theta- \frac {1}{3})+ \frac {1}{2} > 0$ always holds for any $\theta\in[0, \frac {\pi}{2}]$ is $\_\_\_\_\_\_$.

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