
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 function $f(x)$ and its derivative $f''(x)$ on $\mathbb{R}$, and for any real number $x$, it satisfies $f(x)+f(-x)=2x^{2}$, and for $x < 0$, $f''(x)+1 < 2x$, find the minimum value of the real number $a$ such that $f(a+1) \leqslant f(-a)+2a+1$.

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